This is a list with the formulas and proofs in GR.

The table of integrals by Gradshteyn and Ryzhik is one of the most widely used ones.
It is our goal to present proofs of these formulas and put them in some context.

The whole collection of proofs will appear published in SCIENTIA, Series A.

If you want to help, please send the details to vhm at math.tulane.edu

If you experience a problem downloading a formula/proof email me at vhm at math.tulane.edu


Directory of notes on the integrals in GR.

Index of the papers with formulas in GR.

Thanks to those who contribute with a proof:

M. Albano, T. Amdeberhan, E. Beyerstedt, S. Boettner,
K. Boyadzhiev, L. Glasser, X. Guan, L. Jiu,
M. Knight, K. Kohl, C. Koutschan, A. MacLeod,
L. Medina, R. Posey, F. Stan, A. Straub

The notation not yet means that either I have not written the proof or I do not know how to do it.


2.3: The exponential function

Section 2.31 Forms containing $e^{ax}$

Section 2.311


Section 2.312

Section 2.313

Section 2.314


Section 2.315




3.1-3.2: Power and algebraic functions

Section 3.11 Rational functions

Section 3.112 contains 6 formulas.


3.12: Products of rational functions and expresssions that can be reduced to square roots of first and second-degree polynomials

Section 3.121 contains 3 formulas.


3.13-3.17: Expresssions that can be reduced to square roots of third and fourth-degree polynomials
and their products with rational functions.

Section 3.131 contains 8 formulas.


Section 3.132 contains 6 formulas.


Section 3.133 contains 18 formulas.


Section 3.134 contains 18 formulas.


Section 3.135 contains 12 formulas.

Section 3.136 contains 2 formulas.


Section 3.137 contains 8 formulas.

Section 3.138 contains 8 formulas.


Section 3.139 contains 28 formulas.


Section 3.141 contains 36 formulas.


Section 3.142 contains 36 formulas.


Section 3.143 contains 2 formulas.


Section 3.144 contains 12 formulas.


Section 3.145 contains 4 formulas.


Section 3.146 contains 3 formulas.


Section 3.147 contains 8 formulas.


Section 3.148 contains 8 formulas.


Section 3.149 contains 8 formulas.


Section 3.151 contains 8 formulas.


Section 3.152 contains 12 formulas.


Section 3.153 contains 10 formulas.


Section 3.154 contains 9 formulas.


Section 3.155 contains 9 formulas.


Section 3.156 contains 9 formulas.


Section 3.157 contains 12 formulas.


Section 3.158 contains 18 formulas.


Section 3.159 contains 18 formulas.


Section 3.161 contains 9 formulas.


Section 3.162 contains 18 formulas.



Section 3.163 contains 6 formulas.


Section 3.164 contains 8 formulas.


Section 3.165 contains 12 formulas.



Section 3.166 contains 23 formulas.


Section 3.167 contains 32 formulas.



Section 3.168 contains 72 formulas.


Section 3.169 contains 18 formulas.


Section 3.171 contains 18 formulas.


Section 3.172 contains 18 formulas.



Section 3.173 contains 2 formulas.


Section 3.174 contains 4 formulas.


Section 3.175 contains 2 formulas.


Section 3.181 contains 2 formulas.


Section 3.182 contains 2 formulas.


Section 3.183 contains 3 formulas.



Section 3.184 contains 2 formulas.


Section 3.185 contains 7 formulas.


Section 3.186 contains 5 formulas.



3.19-3.23: Combinations of powers of x and powers of binomials of the form a+bx

Section 3.191 contains 3 formulas.


Section 3.192 contains 4 formulas.

Section 3.193 contains a single formula.


Section 3.194 contains 8 formulas.

Section 3.195 contains a single formula.

Section 3.196 contains 5 formulas.

Section 3.197 contains 12 formulas.

Section 3.198 contains a single formula.

Section 3.199 contains a single formula.

Section 3.211 contains a single formula:

Section 3.212 contains a single formula:

Section 3.213 contains a single formula:

Section 3.214 contains a single formula:

Section 3.215 contains a single formula:

Section 3.216 contains 2 formulas.

Section 3.217 contains a single formula.

Section 3.218 contains a single formula.

Section 3.219 contains a single formula.

Section 3.221 contains 2 formulas.

Section 3.222 contains 2 formulas.

Section 3.223 contains 3 formulas:

Section 3.224 contains a single formula.

Section 3.225 contains 3 formulas.

Section 3.226 contains 2 formulas.

Section 3.227 contains 2 formulas:


Section 3.228 contains 6 formulas:

Section 3.229 contains a single formula.

Section 3.231 contains 6 formulas.


Section 3.232 contains a single formula.


Section 3.233 contains a single formula.


Section 3.234 contains 2 formulas.


Section 3.235 contains a single formula.


Section 3.236 contains a single formula:

Section 3.237 contains a single formula:

Section 3.238 contains 3 formulas:


3.24-3.27: Powers of x, binomials of the form .... and of polynomials in x

Section 3.241 contains 6 formulas.

Section 3.242 contains two formulas:

Section 3.243 contains a single formula:

Section 3.244 contains 4 formulas.

Section 3.245 contains a single formula.

Section 3.246 contains a single formula.

Section 3.247 contains 2 formulas.

Section 3.248 contains 6 formulas:

Section 3.249 contains 8 formulas:

Section 3.251 contains 11 formulas.

Section 3.252 contains 14 formulas:


Section 3.253 contains a single formula:

Section 3.254 contains 2 formulas:

Section 3.255 contains a single formula:

Section 3.256 contains 2 formulas:

Section 3.257 contains a single formula:

Section 3.258 contains 7 formulas:


Section 3.259 contains 3 formulas:


Section 3.261 contains 4 formulas:

Section 3.262 contains a single formula:

Section 3.263 contains a single formula:


Section 3.264 contains 7 formulas:

Section 3.265 contains a single formula.


Section 3.266 contains a single formula:


Section 3.267 contains 3 formulas.


Section 3.268 contains 3 formulas:


Section 3.269 contains 3 formulas.


Section 3.271 contains 5 formulas:


Section 3.272 contains 2 formulas.


Section 3.273 contains 4 formulas:


Section 3.274 contains 3 formulas:

Section 3.275 contains 4 formulas:

Section 3.276 contains 2 formulas:

Section 3.277 contains 7 formulas:

Section 3.278 contains a single formula:


3.31: Exponential functions.

Section 3.310 contains a single formula.

Section 3.311 contains 13 formulas.

Section 3.312 contains 3 formulas.

Section 3.313 contains 2 formulas.

Section 3.314 contains a single formula.

Section 3.315 contains 2 formulas:

Section 3.316 contains a single formula.

Section 3.317 contains 2 formulas.

Section 3.318 contains 2 formulas:


3.32-3.34: Exponentials of more complicated arguments.

Section 3.321 contains 7 formulas:


Section 3.322 contains 3 formulas:


Section 3.323 contains 3 formulas:


Section 3.324 contains 2 formulas.


Section 3.325 contains a single formula.


Section 3.326 contains 5 formulas:



Exponentials of exponentials.

Section 3.327 contains a single formula.


Section 3.328 contains a single formula.


Section 3.329 contains a single formula.


Section 3.331 contains 4 formulas.


Section 3.332 contains a single formula:


Section 3.333 contains 3 formulas:


Section 3.334 contains a single formula.


Exponentials of hyperbolic functions.

Section 3.335 contains a single formula:


Section 3.336 contains 3 formulas:


Section 3.337 contains 3 formulas:



Exponentials of trigonometric functions and logarithms.

Section 3.338 contains 5 formulas:


Section 3.339 contains a single formula.


Section 3.341 contains a single formula:

Section 3.342 contains a single formula.


3.35: Combinations of exponentials and rational functions.

Section 3.351 contains 9 formulas.

Section 3.352 contains 7 formulas:



Section 3.353 contains 5 formulas:


Section 3.354 contains 5 formulas:


Section 3.355 contains 4 formulas:


Section 3.356 contains 4 formulas:


Section 3.357 contains 6 formulas:


Section 3.358 contains 8 formulas:


Section 3.359 contains a single formula:



3.36-3.37: Combinations of exponentials and algebraic functions.

Section 3.361 contains 3 formulas.


Section 3.362 contains 2 formulas.


Section 3.363 contains 2 formulas.


Section 3.364 contains 3 formulas.


Section 3.365 contains 2 formulas:


Section 3.366 contains 3 formulas:


Section 3.367 contains a single formula:


Section 3.368 contains a single formula:


Section 3.369 contains a single formula.


Section 3.371 contains a single formula.


Section 3.372 contains a single formula.


Section 3.373 contains a single formula:


Section 3.374 contains two formulas:



3.38-3.39: Combinations of exponentials and arbitrary powers.

Section 3.381 contains 11 formulas:


Section 3.382 contains 7 formulas:


Section 3.383 contains 10 formulas:


Section 3.384 contains 9 formulas:


Section 3.385 contains a single formula:


Section 3.386 contains two formulas:


Section 3.387 contains 7 formulas:


Section 3.388 contains 4 formulas:


Section 3.389 contains 9 formulas:


Section 3.391 contains a single formula:


Section 3.392 contains 4 formulas:


Section 3.393 contains a single formula:


Section 3.394 contains a single formula:


Section 3.395 contains 3 formulas:



3.41-3.44: Combinations of rational functions of powers and exponentials.

Section 3.411 contains 32 formulas.


Section 3.412 contains a single formula:


Section 3.413 contains 3 formulas:


Section 3.414 contains a single formula:


Section 3.415 contains 4 formulas:


Section 3.416 contains 3 formulas:


Section 3.417 contains 2 formulas.


Section 3.418 contains 3 formulas:


Section 3.419 contains 7 formulas:


Section 3.421 contains 5 formulas:


Section 3.422 contains a single formula:


Section 3.423 contains 6 formulas:



Section 3.424 contains 5 formulas:



Section 3.425 contains 2 formulas:



Section 3.426 contains 2 formulas:


Section 3.427 contains 11 formulas:


Section 3.428 contains 7 formulas:



Section 3.429 contains a single formula.


Section 3.431 contains 2 formulas:


Section 3.432 contains 2 formulas:


Section 3.433 contains a single formula.


Section 3.434 contains 2 formulas.


Section 3.435 contains 4 formulas.


Section 3.436 contains a single formula.


Section 3.437 contains a single formula:


Section 3.438 contains 4 formulas:


Section 3.439 contains a single formula:


Section 3.441 contains a single formula:


Section 3.442 contains 3 formulas:


Section 3.443 contains 3 formulas:




3.45: Combinations of powers and algebraic functions of exponentials.

Section 3.451 contains 2 formulas.

Section 3.452 contains 5 formulas.

Section 3.453 contains 2 formulas:

Section 3.454 contains 2 formulas:

Section 3.455 contains 2 formulas:

Section 3.456 contains 2 formulas:

Section 3.457 contains 3 formulas.

Section 3.458 contains 2 formulas:


3.46-3.48: Combinations of exponentials of more complicated arguments and powers.

Section 3.461 contains 9 formulas:

Section 3.462 contains 25 formulas:


Section 3.463 contains a single formula.


Section 3.464 contains a single formula.


Section 3.465 contains a single formula.


Section 3.466 contains 3 formulas.


Section 3.467 contains a single formula.


Section 3.468 contains 2 formulas.


Section 3.469 contains 3 formulas:


Section 3.471 contains 16 formulas:


Section 3.472 contains 5 formulas.


Section 3.473 contains a single formula.


Section 3.474 contains 2 formulas:



Section 3.475 contains 3 formulas.


Section 3.476 contains 2 formulas.


Section 3.477 contains 2 formulas:


Section 3.478 contains 4 formulas:


Section 3.479 contains 2 formulas:


Section 3.481 contains 2 formulas.


Section 3.482 contains 3 formulas:


Section 3.483 contains a single formula:


Section 3.484 contains a single formula.


Section 3.485 contains a single formula:


Section 3.486 contains a single formula:


Section 3.487 contains a single formula:


Section 3.511 contains 10 formulas.


Section 3.512 contains 2 formulas.


Section 3.513 contains 4 formulas:


Section 3.514 contains 4 formulas:


Section 3.515 contains a single formula.


Section 3.516 contains 3 formulas:


Section 3.517 contains 2 formulas:


Section 3.518 contains 6 formulas:


Section 3.519 contains a single formula:


Section 3.521 contains 4 formulas.


Section 3.522 contains 10 formulas.


Section 3.523 contains 12 formulas.


Section 3.524 contains 23 formulas.


Section 3.525 contains 8 formulas:


Section 3.526 contains 3 formulas:


Section 3.527 contains 16 formulas:


Section 3.528 contains 2 formulas:


Section 3.529 contains 3 formulas:


Section 3.531 contains 8 formulas:



Section 3.532 contains 2 formulas:



Section 3.533 contains 4 formulas:


Section 3.534 contains 2 formulas:


Section 3.535 contains a single formula:



Section 3.536 contains 4 formulas:



Section 3.541 contains 10 formulas:


Section 3.542 contains 2 formulas:



Section 3.543 contains 4 formulas:



Section 3.544 contains a single formula:


Section 3.545 contains 2 formulas:


Section 3.546 contains 4 formulas.


Section 3.547 contains 20 formulas:



Section 3.548 contains 2 formulas:



Section 3.549 contains 4 formulas:



Section 3.551 contains 10 formulas:


Section 3.552 contains 6 formulas.



Section 3.553 contains 2 formulas.



Section 3.554 contains 6 formulas:



Section 3.555 contains 2 formulas:



Section 3.556 contains 2 formulas:



Section 3.557 contains 7 formulas:


Section 3.558 contains 7 formulas:


Section 3.559 contains a single formula:


Section 3.561 contains a single formula:


Section 3.562 contains 6 formulas:


Section 3.611 contains 3 formulas:


Section 3.612 contains 7 formulas:


Section 3.613 contains 13 formulas:


Section 3.614 contains a single formula:


Section 3.615 contains 3 formulas:


Section 3.616 contains 8 formulas:


Section 3.617 contains a single formula:


3.62: Powers of trigonometric functions.

Section 3.621 contains 7 formulas:


Section 3.622 contains 4 formulas.


Section 3.623 contains 3 formulas.


Section 3.624 contains 6 formulas:


Section 3.625 contains 4 formulas.


Section 3.626 contains 2 formulas.


Section 3.627 contains a single formula.


Section 3.628 contains a single formula.


Section 3.631 contains 20 formulas:


Section 3.632 contains 5 formulas:



Section 3.633 contains 5 formulas:


Section 3.634 contains 3 formulas:


Section 3.635 contains 3 formulas:


Section 3.636 contains 3 formulas.


Section 3.637 contains 4 formulas:



Section 3.638 contains 3 formulas:


Section 3.641 contains 2 formulas:


Section 3.642 contains 4 formulas:


Section 3.643 contains 2 formulas:


Section 3.644 contains 5 formulas:


Section 3.645 contains a single formula:


Section 3.646 contains 2 formulas:


Section 3.647 contains a single formula:


Section 3.648 contains 2 formulas:


Section 3.649 contains 2 formulas:


Section 3.651 contains 2 formulas:


Section 3.652 contains 3 formulas:


Section 3.653 contains 6 formulas:


Section 3.654 contains 3 formulas:


Section 3.655 contains a single formula:


Section 3.656 contains 2 formulas:


3.66: Forms containing powers of linear functions of trigonometric functions.

Section 3.661 contains 4 formulas:


Section 3.662 contains 5 formulas:


Section 3.663 contains 2 formulas:


Section 3.664 contains 5 formulas:


Section 3.665 contains 2 formulas:


Section 3.666 contains 5 formulas:



Section 3.667 contains 7 formulas:


Section 3.668 contains 2 formulas:


Section 3.669 contains a single formula:


Section 3.670 contains 2 formulas:



Section 3.671 contains 5 formulas:


Section 3.672 contains 2 formulas:


Section 3.673 contains a single formula:



Section 3.674 contains 3 formulas:



Section 3.675 contains 2 formulas:



Section 3.676 contains 3 formulas:



Section 3.677 contains 2 formulas:


Section 3.678 contains 5 formulas:



Section 3.679 contains 3 formulas:


Section 3.681 contains 4 formulas:


Section 3.682 contains a single formula:


Section 3.683 contains 4 formulas:


Section 3.684 contains a single formula:


Section 3.685 contains 7 formulas:


Section 3.686 contains a single formula:


Section 3.687 contains 6 formulas:


Section 3.688 contains 20 formulas:


Section 3.689 contains 2 formulas:



3.69-3.71: Trigonometric functions of more complicated arguments.

Section 3.691 contains 22 formulas:


Section 3.692 contains 6 formulas:


Section 3.693 contains 2 formulas:



Section 3.694 contains 2 formulas:


Section 3.695 contains 2 formulas:


Section 3.696 contains 4 formulas:


Section 3.697 contains a single formula:


Section 3.698 contains 4 formulas:


Section 3.699 contains 5 formulas:


Section 3.711 contains a single formula:


Section 3.712 contains 2 formulas:


Section 3.713 contains 2 formulas:


Section 3.714 contains 5 formulas:



Section 3.715 contains 21 formulas:



Section 3.716 contains 13 formulas:


Section 3.717 contains a single formula:


Section 3.718 contains 7 formulas:


Section 3.719 contains 3 formulas:


Section 3.721 contains 3 formulas:


Section 3.722 contains 8 formulas:


Section 3.723 contains 12 formulas:


Section 3.724 contains 3 formulas:


Section 3.725 contains 3 formulas:


Section 3.726 contains 2 formulas:


Section 3.727 contains 14 formulas:


Section 3.728 contains 9 formulas:


Section 3.729 contains 4 formulas:


Section 3.731 contains 4 formulas:



Section 3.732 contains 4 formulas:


Section 3.733 contains 5 formulas:


Section 3.734 contains 2 formulas:


Section 3.735 contains a single formula:


Section 3.736 contains 6 formulas:



Section 3.737 contains 6 formulas:


Section 3.738 contains 2 formulas:


Section 3.739 contains 4 formulas:


Section 3.741 contains 3 formulas:


Section 3.742 contains 8 formulas:


Section 3.743 contains 5 formulas:


Section 3.744 contains a single formula:


Section 3.745 contains a single formula:


Section 3.746 contains 2 formulas:


Section 3.747 contains 11 formulas:


Section 3.748 contains 3 formulas:


Section 3.749 contains 3 formulas:



3.75: Combinations of trigonometric and algebraic functions.

Section 3.751 contains 4 formulas:


Section 3.752 contains 2 formulas:


Section 3.753 contains 5 formulas:


Section 3.754 contains 3 formulas:


Section 3.755 contains 2 formulas:


Section 3.756 contains 2 formulas:


Section 3.757 contains 2 formulas.


3.76-3.77: Combinations of trigonometric and powers

Section 3.761 contains 12 formulas:



Section 3.762 contains 3 formulas.


Section 3.763 contains 4 formulas.


Section 3.764 contains 2 formulas:


Section 3.765 contains 2 formulas:


Section 3.766 contains 4 formulas:


Section 3.767 contains 4 formulas:


Section 3.768 contains 14 formulas:


Section 3.769 contains 7 formulas.



Section 3.771 contains 13 formulas.



Section 3.772 contains 6 formulas.



Section 3.773 contains 6 formulas.


Section 3.774 contains 6 formulas.


Section 3.775 contains 11 formulas.


Section 3.776 contains 2 formulas:


Section 3.781 contains 2 formulas:


Section 3.782 contains 3 formulas:


Section 3.783 contains 2 formulas:



Section 3.784 contains 9 formulas:


Section 3.785 contains a single formula:


Section 3.786 contains 3 formulas:


Section 3.787 contains 4 formulas:



Section 3.788 contains a single formula:


Section 3.789 contains a single formula:


Section 3.791 contains 12 formulas:



Section 3.792 contains 20 formulas:



Section 3.793 contains 2 formulas:


Section 3.794 contains 6 formulas:


Section 3.795 contains a single formula:


Section 3.796 contains 2 formulas:


Section 3.797 contains 3 formulas:


Section 3.798 contains 2 formulas:


Section 3.799 contains 3 formulas:


Section 3.811 contains 5 formulas:



Section 3.812 contains 15 formulas:



Section 3.813 contains 11 formulas:



Section 3.814 contains 15 formulas:


Section 3.815 contains 4 formulas:


Section 3.816 contains 4 formulas:


Section 3.817 contains 8 formulas:



Section 3.818 contains 9 formulas:



Section 3.819 contains 16 formulas:



3.82-3.83: Powers of trigonometric functions combined with other powers.

Section 3.821 contains 14 formulas:


Section 3.822 contains 2 formulas:


Section 3.823 contains a single formula:


Section 3.824 contains 11 formulas:


Section 3.825 contains 4 formulas:


Section 3.826 contains 2 formulas:


Section 3.827 contains 15 formulas:


Section 3.828 contains 21 formulas:


Section 3.829 contains 2 formulas:


Section 3.831 contains 4 formulas:


Section 3.832 contains 37 formulas:


Section 3.833 contains 2 formulas:


Section 3.834 contains 2 formulas:


Section 3.835 contains 2 formulas:


Section 3.836 contains 6 formulas.


Section 3.837 contains 11 formulas.


Section 3.838 contains 4 formulas:


Section 3.839 contains 10 formulas:



Section 3.841 contains 4 formulas:


Section 3.842 contains 7 formulas:


Section 3.843 contains 4 formulas:

Section 3.844 contains 8 formulas:


Section 3.845 contains 3 formulas:


Section 3.846 contains 8 formulas:



Section 3.847 contains a single formula:


Section 3.848 contains 7 formulas:



Section 3.849 contains 3 formulas:


Section 3.851 contains 5 formulas:



Section 3.852 contains 6 formulas:


Section 3.853 contains 4 formulas:


Section 3.854 contains 4 formulas:


Section 3.855 contains 6 formulas:


Section 3.856 contains 6 formulas:



Section 3.857 contains 2 formulas:


Section 3.858 contains 2 formulas:


Section 3.859 contains a single formula:



Section 3.861 contains 2 formulas:


Section 3.862 contains a single formula:


Section 3.863 contains 2 formulas:


Section 3.864 contains 2 formulas:


Section 3.865 contains 4 formulas:



Section 3.866 contains 3 formulas:


Section 3.867 contains 2 formulas:


Section 3.868 contains 4 formulas:



Section 3.869 contains 2 formulas:


Section 3.871 contains 4 formulas:



Section 3.872 contains 2 formulas:


Section 3.873 contains 2 formulas:



Section 3.874 contains 6 formulas:


Section 3.875 contains 3 formulas:


Section 3.876 contains 8 formulas:


Section 3.877 contains 4 formulas:



Section 3.878 contains 3 formulas:


Section 3.879 contains a single formula:


Section 3.881 contains 17 formulas:



Section 3.882 contains 6 formulas:


Section 3.883 contains 3 formulas:



Section 3.884 contains a single formula:



Section 3.891 contains 2 formulas:



Section 3.892 contains 5 formulas:



Section 3.893 contains 7 formulas:


Section 3.894 contains a single formula:


Section 3.895 contains 16 formulas:



Section 3.896 contains 4 formulas:


Section 3.897 contains 2 formulas:


Section 3.898 contains 3 formulas:


Section 3.899 contains 3 formulas:



Section 3.911 contains 6 formulas:


Section 3.912 contains 2 formulas:


Section 3.913 contains 2 formulas:


Section 3.914 contains 10 formulas:


Section 3.915 contains 5 formulas:


Section 3.916 contains 5 formulas:


Section 3.917 contains 2 formulas:


Section 3.918 contains 3 formulas:


Section 3.919 contains 2 formulas:


Section 3.921 contains 3 formulas:


Section 3.922 contains 4 formulas:


Section 3.923 contains 2 formulas:


Section 3.924 contains 2 formulas:


Section 3.925 contains 2 formulas:


Section 3.926 contains 2 formulas:


Section 3.927 contains a single formula:


Section 3.928 contains 2 formulas:


Section 3.929 contains a single formula:


Section 3.931 contains 4 formulas:


Section 3.932 contains 2 formulas:


Section 3.933 contains a single formula:


Section 3.934 contains 2 formulas:


Section 3.935 contains a single formula:


Section 3.936 contains 5 formulas:


Section 3.937 contains 4 formulas:


Section 3.938 contains 2 formulas:



Section 3.939 contains 3 formulas:


Section 3.941 contains 2 formulas:


Section 3.942 contains 2 formulas:


Section 3.943 contains a single formula:


Section 3.944 contains 14 formulas:


Section 3.945 contains 3 formulas:



Section 3.946 contains 2 formulas:


Section 3.947 contains 3 formulas:


Section 3.948 contains 5 formulas:



Section 3.949 contains 5 formulas:


Section 3.951 contains 21 formulas:


Section 3.952 contains 10 formulas:



Section 3.953 contains 4 formulas:



Section 3.954 contains 2 formulas:



Section 3.955 contains a single formula:


Section 3.956 contains a single formula:


Section 3.957 contains 2 formulas:



Section 3.958 contains 2 formulas:



Section 3.959 contains a single formula:



Section 3.961 contains 2 formulas:



Section 3.962 contains 2 formulas:


Section 3.963 contains 4 formulas:



Section 3.964 contains 3 formulas:



Section 3.965 contains 2 formulas:



Section 3.966 contains 6 formulas:



Section 3.967 contains 3 formulas:



Section 3.968 contains 2 formulas:



Section 3.969 contains 2 formulas:



Section 3.971 contains 2 formulas:



Section 3.972 contains 2 formulas:



Section 3.973 contains 6 formulas:



Section 3.974 contains 10 formulas:



Section 3.975 contains 2 formulas:


Section 3.976 contains a single formula:



3.98-3.99: Combinations of trigonometric and hyperbolic functions.

Section 3.981 contains 12 formulas:


Section 3.982 contains 3 formulas:


Section 3.983 contains 7 formulas:



Section 3.984 contains 7 formulas:


Section 3.985 contains 3 formulas:



Section 3.986 contains 4 formulas:



Section 3.987 contains 2 formulas:



Section 3.988 contains 3 formulas:



Section 3.989 contains 6 formulas:



Section 3.991 contains 2 formulas:


Section 3.992 contains 2 formulas:


Section 3.993 contains a single formula:


Section 3.994 contains 7 formulas:


Section 3.995 contains 2 formulas:


Section 3.996 contains 5 formulas:


Section 3.997 contains 4 formulas:


Section 4.111 contains 7 formulas:


Section 4.112 contains 2 formulas:


Section 4.113 contains 13 formulas.


Section 4.114 contains 2 formulas:


Section 4.115 contains 14 formulas:


Section 4.116 contains 2 formulas:


Section 4.117 contains 9 formulas:


Section 4.118 contains a single formula:


Section 4.119 contains a single formula:


Section 4.121 contains 2 formulas:


Section 4.122 contains 2 formulas:


Section 4.123 contains 7 formulas:

Section 4.124 contains 2 formulas.


Section 4.125 contains 3 formulas.


Section 4.126 contains 4 formulas.



Section 4.131 contains 4 formulas.



Section 4.132 contains 5 formulas.



Section 4.133 contains 2 formulas.


Section 4.134 contains 2 formulas.



Section 4.135 contains 2 formulas.



Section 4.136 contains 4 formulas.


Section 4.137 contains 4 formulas.


Section 4.138 contains 4 formulas.


Section 4.141 contains 4 formulas.



Section 4.142 contains 4 formulas.


Section 4.143 contains 2 formulas.


Section 4.144 contains a single formula.


Section 4.145 contains 2 formulas.


Section 4.146 contains 6 formulas.



4.21: Logarithmic functions.

Section 4.211 contains 2 formulas.


Section 4.212 contains 9 formulas.


Section 4.213 contains 8 formulas.


Section 4.214 contains 4 formulas.


Section 4.215 contains 4 formulas.


Section 4.216 contains 2 formulas.



4.22: Logarithms of more complicated arguments.

Section 4.221 contains 3 formulas.


Section 4.222 contains 8 formulas:

Section 4.223 contains 3 formulas.


Section 4.224 contains 15 formulas:


Section 4.225 contains 4 formulas:


Section 4.226 contains 7 formulas:

Section 4.227 contains 17 formulas.


Section 4.228 contains 8 formulas.


Section 4.229 contains 7 formulas:



4.23: Combinations of logarithms and rational functions.

Section 4.231 contains 20 formulas:


Section 4.232 contains 3 formulas:


Section 4.233 contains 5 formulas:


Section 4.234 contains 8 formulas.



Section 4.235 contains 4 formulas.


Section 4.236 contains 2 formulas.



4.24: Combinations of logarithms and algebraic functions.

Section 4.241 contains 11 formulas.


Section 4.242 contains 6 formulas:


Section 4.243 contains a single formula.


Section 4.244 contains 3 formulas.


Section 4.245 contains 2 formulas.


Section 4.246 contains a single formula.


Section 4.247 contains 2 formulas.


4.25: Combinations of logarithms and powers.

Section 4.251 contains 6 formulas.


Section 4.252 contains 4 formulas.


Section 4.253 contains 7 formulas.


Section 4.254 contains 6 formulas.


Section 4.255 contains 3 formulas.


Section 4.256 contains a single formula.


Section 4.257 contains 5 formulas:



4.26-4.27: Combinations involving powers of the logarithm and other powers.

Section 4.261 contains 21 formulas:


Section 4.262 contains 9 formulas.


Section 4.263 contains 3 formulas:


Section 4.264 contains 3 formulas.


Section 4.265 contains a single formula.


Section 4.266 contains 2 formulas.


Section 4.267 contains 43 formulas:


Section 4.268 contains 7 formulas:


Section 4.269 contains 7 formulas:


Section 4.271 contains 16 formulas.


Section 4.272 contains 19 formulas:


Section 4.273 contains a single formula.


Section 4.274 contains a single formula.


Section 4.275 contains 2 formulas.


4.28: Combinations of rational functions of ln x and powers.

Section 4.281 contains 7 formulas:


Section 4.282 contains 13 formulas:


Section 4.283 contains 14 formulas:


Section 4.284 contains 2 formulas:


Section 4.285 contains a single formula:



4.29-4.32: Combinations of logarithmic functions of more complicated arguments and powers.

Section 4.291 contains 30 formulas:

Section 4.292 contains 5 formulas:

Section 4.293 contains 14 formulas.

Section 4.294 contains 13 formulas:

Section 4.295 contains 41 formulas:


Section 4.296 contains 4 formulas:


Section 4.297 contains 10 formulas:


Section 4.298 contains 23 formulas:


Section 4.299 contains 4 formulas:


Section 4.311 contains 6 formulas:



Section 4.312 contains 2 formulas:



Section 4.313 contains 7 formulas:


Section 4.314 contains 4 formulas:


Section 4.315 contains 4 formulas:



Section 4.316 contains 2 formulas:


Section 4.317 contains 11 formulas:


Section 4.318 contains 2 formulas:


Section 4.319 contains 3 formulas:


Section 4.321 contains 2 formulas:


Section 4.322 contains 10 formulas:


Section 4.323 contains 3 formulas:


Section 4.324 contains 3 formulas:


Section 4.325 contains 12 formulas:


Section 4.326 contains 2 formulas:


Section 4.327 contains 3 formulas:



4.33-4.34: Combinations of logarithms and exponentials.

Section 4.331 contains 3 formulas.


Section 4.332 contains 2 formulas:


Section 4.333 contains a single formula.


Section 4.334 contains a single formula:


Section 4.335 contains 3 formulas:

Section 4.336 contains 2 formulas:


Section 4.337 contains 5 formulas:


Section 4.338 contains 2 formulas:


Section 4.339 contains a single formula:


Section 4.341 contains a single formula:


Section 4.342 contains 3 formulas:


Section 4.343 contains a single formula:


Section 4.351 contains 3 formulas:


Section 4.352 contains 4 formulas:


Section 4.353 contains 3 formulas:


Section 4.354 contains 5 formulas:


Section 4.355 contains 4 formulas.


Section 4.356 contains 11 formulas:


Section 4.357 contains 3 formulas:




Section 4.358 contains 5 formulas:


Section 4.359 contains 2 formulas:


Section 4.361 contains 2 formulas:


Section 4.362 contains 2 formulas:


Section 4.363 contains 2 formulas:


Section 4.364 contains 2 formulas:


Section 4.365 contains a single formula:


Section 4.366 contains 3 formulas:



Section 4.367 contains a single formulas:


Section 4.368 contains a single formula:


Section 4.369 contains 2 formulas:



Section 4.371 contains 3 formulas:


Section 4.372 contains 2 formulas:


Section 4.373 contains 6 formulas:


Section 4.374 contains 2 formulas:


Section 4.375 contains 2 formulas:


Section 4.376 contains 12 formulas:


Section 4.377 contains a single formula:


Section 4.381 contains 4 formulas:

Section 4.382 contains 5 formulas:

Section 4.383 contains 2 formulas:


Section 4.384 contains 26 formulas:



Section 4.385 contains 5 formulas:


Section 4.386 contains 4 formulas:


Section 4.387 contains 10 formulas:


Section 4.388 contains 6 formulas:


Section 4.389 contains 6 formulas:


Section 4.391 contains 4 formulas:


Section 4.392 contains 2 formulas:


Section 4.393 contains 5 formulas:


Section 4.394 contains 9 formulas:


Section 4.395 contains 2 formulas:


Section 4.396 contains 7 formulas:


Section 4.397 contains 16 formulas:


Section 4.398 contains 3 formulas:


Section 4.399 contains 3 formulas:


Section 4.411 contains 3 formulas:


Section 4.412 contains 5 formulas:


Section 4.413 contains 5 formulas:


Section 4.414 contains 10 formulas:


Section 4.415 contains 2 formulas:


Section 4.416 contains 4 formulas:


4.42-4.43: Combinations of logarithms, trigonometric functions, and powers.

Section 4.421 contains 5 formulas:


Section 4.422 contains 2 formulas:


Section 4.423 contains 3 formulas:


Section 4.424 contains 2 formulas:


Section 4.425 contains 4 formulas:


Section 4.426 contains 2 formulas:


Section 4.427 contains a single formula:


Section 4.428 contains 4 formulas:

Section 4.429 contains a single formula:

Section 4.431 contains 4 formulas:



Section 4.432 contains 26 formulas:


Section 4.441 contains 2 formulas:


Section 4.442 contains a single formula:


4.51: Inverse Trigonometric Functions.

Section 4.511 contains a single formula:


Section 4.512 contains a single formula:


4.52: Combinations of arcsines, arccosines, and powers.

Section 4.521 contains 8 formulas:


Section 4.522 contains 10 formulas:


Section 4.523 contains 6 formulas:

Section 4.524 contains 2 formulas:


4.53-4.54: Combinations of arctangents, arccotangents, and powers.

Section 4.531 contains 13 formulas:


Section 4.532 contains 5 formulas:


Section 4.533 contains 4 formulas:


Section 4.534 contains a single formula:


Section 4.535 contains 15 formulas:


Section 4.536 contains 3 formulas:



Section 4.537 contains 5 formulas:


Section 4.538 contains 2 formulas:


Section 4.539 contains a single formula:

Section 4.541 contains a single formula:


Section 4.551 contains 4 formulas:


Section 4.552 contains a single formula:


Section 4.553 contains a single formula:


Section 4.561 contains a single formula:


Section 4.571 contains a single formula:

Section 4.572 contains a single formula:


Section 4.573 contains 4 formulas:


Section 4.574 contains 4 formulas:



Section 4.575 contains 3 formulas:


Section 4.576 contains 2 formulas:


Section 4.577 contains 2 formulas:



Section 4.581 contains a single formula:


Section 4.591 contains 2 formulas:


Section 4.592 contains a single formula:



Section 4.593 contains 2 formulas:



Section 4.594 contains a single formula:



5.1: Elliptic integrals and functions
5.11: Complete elliptic integrals

Section 5.111 contains 2 formulas:


Section 5.112 contains 13 formulas:


Section 5.113 contains 6 formulas:

Section 5.114 contains a single formula:


Section 5.115 contains 3 formulas:

Section 5.121 contains a single formula:


Section 5.122 contains a single formula:


Section 5.123 contains 2 formulas:



Section 5.124 contains 11 formulas:



Section 5.125 contains 2 formulas:


Section 5.126 contains a single formula:


Section 5.127 contains a single formula:


Section 5.128 contains a single formula:



5.13: Jacobian elliptic functions.

Section 5.131 contains 3 formulas:


Section 5.132 contains 3 formulas:


Section 5.133 contains 3 formulas:


Section 5.134 contains 3 formulas:


Section 5.135 contains 6 formulas:


Section 5.136 contains 3 formulas:


Section 5.137 contains 6 formulas:


Section 5.138 contains 3 formulas:


Section 5.139 contains 3 formulas:


Section 5.141 contains 5 formulas:



5.2: The exponential integral function.
5.21: The exponential integral function.

Section 5.211 contains a single formula:


Section 5.221 contains 10 formulas:


Section 5.231 contains 2 formulas:


Section 5.31 contains 2 formulas:



Section 5.32 contains 2 formulas:



Section 5.33 contains 3 formulas:


Section 5.34 contains 2 formulas:


Section 5.41 contains a single formula:


Section 5.42 contains a single formula:


Section 5.43 contains a single formula:



Section 5.51 contains a single formula:


Section 5.52 contains 2 formulas:


Section 5.53 contains a single formula:


Section 5.54 contains 3 formulas:


Section 5.55 contains a single formula:


Section 5.56 contains 2 formulas:



6.1: Elliptic integrals and functions.

6.11 Forms containing F(x,k)

Section 6.111 contains a single formula:


Section 6.112 contains 3 formulas:


Section 6.113 contains 2 formulas:


Section 6.114 contains a single formula:


Section 6.115 contains a single formula:


Section 6.121 contains a single formula:


Section 6.122 contains a single formula:


Section 6.123 contains a single formula:


Section 6.124 contains a single formula:


Section 6.131 contains a single formula:


Section 6.132 contains a single formula:


Section 6.133 contains a single formula:


Section 6.141 contains 2 formulas:


Section 6.142 contains a single formula:


Section 6.143 contains a single formula:


Section 6.144 contains a single formula:


Section 6.145 contains a single formula:


Section 6.146 contains a single formula:


Section 6.147 contains a single formula:


Section 6.148 contains 3 formulas:


Section 6.149 contains 4 formulas:


Section 6.151 contains a single formula:


Section 6.152 contains a single formula:


Section 6.153 contains a single formula:


Section 6.154 contains a single formula:


Section 6.161 contains 4 formulas:


Section 6.162 contains 4 formulas:


Section 6.163 contains 2 formulas:


Section 6.164 contains a single formula:


Section 6.165 contains a single formula:


Section 6.17 contains 2 formulas:



6.2-6.31: The exponential integral and functions generated by it.


6.21: The logarithmic integral.

Section 6.211 contains a single formula:


Section 6.212 contains 4 formulas:


Section 6.213 contains 12 formulas:


Section 6.214 contains 2 formulas:


Section 6.215 contains 2 formulas:


Section 6.216 contains 2 formulas:


6.22-6.23: The exponential integral function.

Section 6.221 contains a single formula:

Section 6.222 contains a single formula:


Section 6.223 contains a single formula:


Section 6.224 contains 2 formulas:


Section 6.225 contains 2 formulas:


Section 6.226 contains 4 formulas:


Section 6.227 contains 2 formulas:


Section 6.228 contains 2 formulas:


Section 6.229 contains a single formula:


Section 6.231 contains a single formula:


Section 6.232 contains 2 formulas:


Section 6.233 contains 2 formulas:


Section 6.234 contains a single formula:


Section 6.241 contains 3 formulas:


Section 6.242 contains a single formula:


Section 6.243 contains 2 formulas:


Section 6.244 contains 2 formulas:


Section 6.245 contains 2 formulas:


Section 6.246 contains 2 formulas:


Section 6.247 contains 2 formulas:


Section 6.248 contains 2 formulas:


Section 6.249 contains a single formula:


Section 6.251 contains 2 formulas:


Section 6.252 contains 4 formulas:


Section 6.253 contains a single formula:



Section 6.254 contains 2 formulas:


Section 6.255 contains 2 formulas:


Section 6.256 contains 4 formulas:


Section 6.257 contains a single formula:


Section 6.258 contains 2 formulas:



Section 6.259 contains 5 formulas:



Section 6.261 contains 2 formulas:


Section 6.262 contains 3 formulas:


Section 6.263 contains 3 formulas:


Section 6.264 contains 2 formulas:


Section 6.271 contains 2 formulas:


Section 6.272 contains a single formula:


Section 6.273 contains 2 formulas:


Section 6.274 contains a single formula:


Section 6.275 contains a single formula:


Section 6.276 contains a single formula:


Section 6.277 contains 2 formulas:




6.28 - 6.31: The probability integral.

Section 6.281 contains 2 formulas:


Section 6.282 contains 2 formulas:


Section 6.283 contains 2 formulas:


Section 6.284 contains a single formula:


Section 6.285 contains 2 formulas:


Section 6.286 contains 2 formulas:


Section 6.287 contains 3 formulas:


Section 6.288 contains a single formula:


Section 6.289 contains 3 formulas:


Section 6.291 contains a single formula:


Section 6.292 contains a single formula:


Section 6.293 contains a single formula:


Section 6.294 contains 2 formulas:


Section 6.295 contains 3 formulas:


Section 6.296 contains a single formula:


Section 6.297 contains 2 formulas:


Section 6.298 contains a single formula:


Section 6.299 contains a single formula:


Section 6.311 contains a single formula:


Section 6.312 contains a single formula:


Section 6.313 contains 2 formulas:


Section 6.314 contains 2 formulas:


Section 6.315 contains 5 formulas:


Section 6.316 contains a single formula:


Section 6.317 contains a single formula:


Section 6.318 contains a single formula:


Section 6.321 contains 2 formulas:


Section 6.322 contains 2 formulas:


Section 6.323 contains 2 formulas:


Section 6.324 contains 2 formulas:


Section 6.325 contains 2 formulas:


Section 6.326 contains 2 formulas:


Section 6.411 contains a single formula:


Section 6.412 contains a single formula:


Section 6.413 contains 2 formulas:


Section 6.414 contains 6 formulas:


Section 6.415 contains 2 formulas:


Section 6.421 contains 4 formulas:


Section 6.422 contains 19 formulas:


Section 6.423 contains 4 formulas:


Section 6.424 contains a single formula:


Section 6.431 contains 2 formulas:


Section 6.432 contains a single formula:


Section 6.433 contains 2 formulas:


Section 6.441 contains 5 formulas:


Section 6.442 contains a single formula:


Section 6.443 contains 6 formulas:


Section 6.451 contains 2 formulas:


Section 6.452 contains 2 formulas:


Section 6.453 contains a single formula:


Section 6.454 contains a single formula:


Section 6.455 contains 2 formulas:


Section 6.456 contains 2 formulas:


Section 6.457 contains 2 formulas:


Section 6.458 contains a single formula:


Section 6.461 contains a single formula:


Section 6.462 contains a single formula:


Section 6.463 contains a single formula:


Section 6.464 contains a single formula:



Section 6.465 contains 2 formulas:


Section 6.466 contains a single formula:


Section 6.467 contains 2 formulas:


Section 6.468 contains a single formula:


Section 6.469 contains 2 formulas:


Section 6.471 contains 3 formulas:


Section 6.472 contains 2 formulas:


Section 6.473 contains a single formula:


Section 6.511 contains 13 formulas:


Section 6.512 contains 10 formulas:


Section 6.513 contains 9 formulas:


Section 6.514 contains 8 formulas:


Section 6.515 contains 3 formulas:


Section 6.516 contains 10 formulas:


Section 6.517 contains a single formula:


Section 6.518 contains a single formula:


Section 6.519 contains 2 formulas:


Section 6.521 contains 15 formulas:



Section 6.522 contains 18 formulas:


Section 6.523 contains a single formula:


Section 6.524 contains 2 formulas:


Section 6.525 contains 3 formulas:


Section 6.526 contains 8 formulas:

Section 6.527 contains 3 formulas:


Section 6.528 contains a single formula:


Section 6.529 contains 2 formulas:


Section 6.531 contains 3 formulas:


Section 6.532 contains 6 formulas:


Section 6.533 contains 5 formulas:



Section 6.534 contains a single formula:


Section 6.535 contains a single formula:


Section 6.536 contains a single formula:


Section 6.537 contains a single formula:


Section 6.538 contains 2 formulas:


Section 6.539 contains 3 formulas:


Section 6.541 contains 3 formulas:


Section 6.542 contains a single formula:


Section 6.543 contains a single formula:


Section 6.544 contains 8 formulas:


Section 6.551 contains 2 formulas:


Section 6.552 contains 7 formulas:


Section 6.553 contains a single formula:



Section 6.554 contains 5 formulas:


Section 6.555 contains a single formula:


Section 6.556 contains a single formula:


Section 6.561 contains 19 formulas:


Section 6.562 contains 3 formulas:


Section 6.563 contains a single formula:


Section 6.564 contains 2 formulas:


Section 6.565 contains 8 formulas:


Section 6.566 contains 5 formulas:


Section 6.567 contains 17 formulas:


Section 6.568 contains 2 formulas:


Section 6.569 contains a single formula:


Section 6.571 contains 3 formulas:



Section 6.572 contains 3 formulas:


Section 6.573 contains 2 formulas:


Section 6.574 contains 3 formulas:


Section 6.575 contains 2 formulas:


Section 6.576 contains 7 formulas:


Section 6.577 contains 2 formulas:


Section 6.578 contains 16 formulas:


Section 6.579 contains 4 formulas:


Section 6.581 contains 5 formulas:


Section 6.582 contains a single formula:


Section 6.583 contains a single formula:


Section 6.584 contains 5 formulas:


Section 6.591 contains 8 formulas:


Section 6.592 contains 17 formulas:


Section 6.593 contains 2 formulas:


Section 6.594 contains 3 formulas:


Section 6.595 contains 2 formulas:


Section 6.596 contains 12 formulas:


Section 6.597 contains a single formula:


Section 6.598 contains a single formula:


Section 6.611 contains 10 formulas:


Section 6.612 contains 6 formulas:


Section 6.613 contains a single formula:


Section 6.614 contains 5 formulas:


Section 6.615 contains a single formula:


Section 6.616 contains 6 formulas:


Section 6.617 contains 2 formulas:


Section 6.618 contains 5 formulas:


Section 6.621 contains 19 formulas:



Section 6.622 contains 3 formulas:


Section 6.623 contains 3 formulas:


Section 6.624 contains 7 formulas:


Section 6.625 contains 11 formulas:


Section 6.626 contains 5 formulas:


Section 6.627 contains a single formula:


Section 6.628 contains 7 formulas:


Section 6.629 contains a single formula:


Section 6.631 contains 10 formulas:


Section 6.632 contains a single formula:


Section 6.633 contains 5 formulas:


Section 6.634 contains a single formula:


Section 6.635 contains 3 formulas:


Section 6.636 contains a single formula:


Section 6.637 contains 3 formulas:


Section 6.641 contains a single formula:


Section 6.642 contains 2 formulas:


Section 6.643 contains 6 formulas:


Section 6.644 contains a single formula:


Section 6.645 contains 3 formulas:


Section 6.646 contains 3 formulas:


Section 6.647 contains 3 formulas:


Section 6.648 contains a single formula:


Section 6.649 contains 4 formulas:


Section 6.651 contains 9 formulas:


Section 6.652 contains a single formula:


Section 6.653 contains 2 formulas:


Section 6.654 contains a single formula:


Section 6.655 contains a single formula:


Section 6.656 contains 2 formulas:


Section 6.661 contains 2 formulas:


Section 6.662 contains 2 formulas:


Section 6.663 contains 5 formulas:


Section 6.664 contains 6 formulas:


Section 6.665 contains a single formula:


Section 6.666 contains a single formula:


Section 6.667 contains 2 formulas:


Section 6.668 contains 2 formulas:


Section 6.669 contains 6 formulas:


Section 6.671 contains 14 formulas:


Section 6.672 contains 10 formulas:


Section 6.673 contains 3 formulas:


Section 6.674 contains 8 formulas:


Section 6.675 contains 4 formulas:


Section 6.676 contains 6 formulas:


Section 6.677 contains 9 formulas:


Section 6.678 contains a single formula:


Section 6.679 contains 7 formulas:


Section 6.681 contains 13 formulas:


Section 6.682 contains 2 formulas:


Section 6.683 contains 10 formulas:


Section 6.684 contains 2 formulas:


Section 6.685 contains a single formula:


Section 6.686 contains 7 formulas:


Section 6.687 contains a single formula:


Section 6.688 contains 3 formulas:


Section 6.691 contains a single formula:


Section 6.692 contains 2 formulas:


Section 6.693 contains 10 formulas:


Section 6.694 contains a single formula:


Section 6.695 contains 4 formulas:


Section 6.696 contains a single formula:


Section 6.697 contains 7 formulas:


Section 6.698 contains 4 formulas:


Section 6.699 contains 14 formulas:


Section 6.711 contains 7 formulas:


Section 6.712 contains 4 formulas:


Section 6.713 contains 3 formulas:


Section 6.714 contains 2 formulas:


Section 6.715 contains 2 formulas:


Section 6.716 contains 2 formulas:


Section 6.717 contains a single formula:


Section 6.718 contains 4 formulas:


Section 6.719 contains 2 formulas:


Section 6.721 contains 6 formulas:


Section 6.722 contains 4 formulas:


Section 6.723 contains a single formula:


Section 6.724 contains 2 formulas:


Section 6.725 contains 4 formulas:


Section 6.726 contains 5 formulas:


Section 6.727 contains 4 formulas:


Section 6.728 contains 6 formulas:


Section 6.729 contains 2 formulas:


Section 6.731 contains 2 formulas:


Section 6.732 contains a single formula:


Section 6.733 contains 4 formulas:


Section 6.734 contains a single formula:


Section 6.735 contains 4 formulas:


Section 6.736 contains 6 formulas:


Section 6.737 contains 6 formulas:


Section 6.738 contains 2 formulas:


Section 6.739 contains a single formula:


Section 6.741 contains 3 formulas:


Section 6.751 contains 3 formulas:


Section 6.752 contains 5 formulas:


Section 6.753 contains 6 formulas:


Section 6.754 contains 3 formulas:


Section 6.755 contains 9 formulas:


Section 6.756 contains 4 formulas:


Section 6.757 contains 2 formulas:


Section 6.758 contains a single formula:


Section 6.761 contains a single formula:


Section 6.762 contains a single formula:


Section 6.763 contains a single formula:


Section 6.771 contains a single formula:


Section 6.772 contains 4 formulas:


Section 6.773 contains a single formula:


Section 6.774 contains a single formula:


Section 6.775 contains a single formula:


Section 6.776 contains a single formula:


Section 6.777 contains a single formula:


Section 6.781 contains a single formula:


Section 6.782 contains 7 formulas:


Section 6.783 contains 4 formulas:


Section 6.784 contains 2 formulas:


Section 6.785 contains a single formula:


Section 6.786 contains a single formula:


Section 6.787 contains a single formula:


Section 6.791 contains 3 formulas:


Section 6.792 contains 5 formulas:


Section 6.793 contains 3 formulas:


Section 6.794 contains 13 formulas:


Section 6.795 contains 5 formulas:


Section 6.796 contains 5 formulas:


Section 6.797 contains 7 formulas:


Section 6.811 contains 3 formulas:


Section 6.812 contains 2 formulas:


Section 6.813 contains 5 formulas:


Section 6.814 contains a single formula:


Section 6.815 contains 2 formulas:


Section 6.821 contains 4 formulas:


Section 6.822 contains a single formula:


Section 6.823 contains 2 formulas:


Section 6.824 contains 2 formulas:


Section 6.825 contains a single formula:


Section 6.831 contains a single formula:


Section 6.832 contains a single formula:


Section 6.841 contains a single formula:


Section 6.842 contains a single formula:


Section 6.843 contains 2 formulas:


Section 6.844 contains a single formula:


Section 6.845 contains 2 formulas:


Section 6.846 contains a single formula:


Section 6.847 contains a single formula:


Section 6.848 contains 2 formulas:


Section 6.849 contains 2 formulas:


Section 6.851 contains 2 formulas:


Section 6.852 contains 3 formulas:


Section 6.853 contains 6 formulas:


Section 6.854 contains 2 formulas:


Section 6.855 contains 3 formulas:


Section 6.856 contains a single formula:


Section 6.857 contains 2 formulas:


Section 6.861 contains a single formula:


Section 6.862 contains 2 formulas:


Section 6.863 contains a single formula:


Section 6.864 contains a single formula:


Section 6.865 contains a single formula:


Section 6.866 contains 2 formulas:


Section 6.867 contains 2 formulas:


Section 6.868 contains a single formula:


Section 6.869 contains 3 formulas:


Section 6.871 contains 2 formulas:


Section 6.872 contains 7 formulas:


Section 6.873 contains a single formula:


Section 6.874 contains 4 formulas:


Section 6.875 contains 2 formulas:


Section 6.876 contains 2 formulas:


Section 6.911 contains 7 formulas:


Section 6.921 contains 5 formulas:


Section 6.922 contains 8 formulas:


Section 6.923 contains 8 formulas:


Section 6.924 contains 6 formulas:


Section 6.925 contains 8 formulas:


Section 6.926 contains a single formula:


Section 6.931 contains 2 formulas:


Section 6.941 contains 10 formulas:




7.1-7.2 Associated Legendre functions .
7.11 Associated Legendre functions

Section 7.111 contains a single formula:


Section 7.112 contains 5 formulas:


Section 7.113 contains 3 formulas:


Section 7.114 contains 3 formulas:


Section 7.115 contains a single formula:


Section 7.121 contains a single formula:


Section 7.122 contains 3 formulas:


Section 7.123 contains a single formula:


Section 7.124 contains a single formula:


Section 7.125 contains a single formula:


Section 7.126 contains 4 formulas:


Section 7.127 contains a single formula:


Section 7.128 contains 2 formulas:


Section 7.129 contains a single formula:


Section 7.131 contains 2 formulas:


Section 7.132 contains 7 formulas:


Section 7.133 contains 2 formulas:


Section 7.134 contains 2 formulas:


Section 7.135 contains 3 formulas:


Section 7.136 contains 3 formulas:


Section 7.137 contains 12 formulas:


Section 7.138 contains a single formula:


Section 7.141 contains 5 formulas:


Section 7.142 contains a single formula:


Section 7.143 contains 2 formulas:


Section 7.144 contains 2 formulas:


Section 7.145 contains 3 formulas:


Section 7.146 contains 3 formulas:


Section 7.147 contains a single formula:


Section 7.148 contains a single formula:


Section 7.149 contains a single formula:


Section 7.151 contains 2 formulas:


Section 7.152 contains a single formula:


Section 7.161 contains 3 formulas:


Section 7.162 contains 7 formulas:


Section 7.163 contains 2 formulas:


Section 7.164 contains 8 formulas:


Section 7.165 contains a single formula:


Section 7.166 contains a single formula:


Section 7.167 contains a single formula:


Section 7.171 contains a single formula:


Section 7.181 contains 2 formulas:


Section 7.182 contains 17 formulas:


Section 7.183 contains a single formula.



Section 7.184 contains 7 formulas:


Section 7.185 contains a single formula.



Section 7.186 contains a single formula.




Section 7.187 contains 10 formulas:


Section 7.188 contains 3 formulas:


Section 7.189 contains 5 formulas:


Section 7.191 contains 2 formulas:


Section 7.192 contains 2 formulas:


Section 7.193 contains 2 formulas:


Section 7.211 contains 2 formulas:


Section 7.212 contains a single formula.


Section 7.213 contains a single formula.


Section 7.214 contains a single formula.


Section 7.215 contains a single formula.


Section 7.216 contains a single formula.


Section 7.217 contains 4 formulas:


Section 7.221 contains 3 formulas:


Section 7.222 contains 5 formulas:


Section 7.223 contains a single formula.


Section 7.224 contains 8 formulas:


Section 7.225 contains 5 formulas:


Section 7.226 contains 3 formulas:


Section 7.227 contains a single formula.


Section 7.228 contains a single formula.


Section 7.231 contains 2 formulas:


Section 7.232 contains 4 formulas:


Section 7.233 contains a single formula.


Section 7.241 contains a single formula.


Section 7.242 contains a single formula.


Section 7.243 contains 5 formulas:


Section 7.244 contains 2 formulas:


Section 7.245 contains 4 formulas:


Section 7.246 contains a single formula.


Section 7.247 contains a single formula.


Section 7.248 contains 2 formulas:


Section 7.249 contains 2 formulas:


Section 7.251 contains 7 formulas:


Section 7.252 contains a single formula.


Section 7.253 contains a single formula.


Section 7.254 contains a single formula.


Section 7.311 contains 4 formulas:


Section 7.312 contains 3 formulas:


Section 7.313 contains 2 formulas:


Section 7.314 contains 7 formulas:


Section 7.315 contains a single formula.


Section 7.316 contains a single formula.


Section 7.317 contains 2 formulas:


Section 7.318 contains a single formula.


Section 7.319 contains 2 formulas:


Section 7.321 contains a single formula.


Section 7.322 contains a single formula.


Section 7.323 contains 2 formulas:


Section 7.324 contains 2 formulas:


Section 7.325 contains 2 formulas:



Section 7.331 contains a single formula.


Section 7.332 contains 2 formulas:


Section 7.333 contains 2 formulas:


Section 7.334 contains 2 formulas:


Section 7.335 contains a single formula.


Section 7.336 contains a single formula.


Section 7.341 contains a single formula.


Section 7.342 contains a single formula.


Section 7.343 contains 2 formulas:


Section 7.344 contains 2 formulas:


Section 7.345 contains 7 formulas:


Section 7.346 contains a single formula.


Section 7.347 contains 2 formulas:


Section 7.348 contains a single formula.


Section 7.349 contains a single formula.


Section 7.351 contains a single formula.


Section 7.352 contains 2 formulas:


Section 7.353 contains 2 formulas:


Section 7.354 contains 4 formulas:


Section 7.355 contains 2 formulas:


Section 7.361 contains a single formula.


Section 7.362 contains a single formula.


Section 7.371 contains a single formula.


Section 7.372 contains a single formula.


Section 7.373 contains 2 formulas:


Section 7.374 contains 10 formulas:


Section 7.375 contains 2 formulas:


Section 7.376 contains 3 formulas:


Section 7.377 contains a single formula.


Section 7.378 contains a single formula.


Section 7.379 contains 2 formulas:


Section 7.381 contains a single formula.


Section 7.382 contains a single formula.


Section 7.383 contains 5 formulas:


Section 7.384 contains a single formula.


Section 7.385 contains 2 formulas:


Section 7.386 contains a single formula.


Section 7.387 contains 2 formulas:


Section 7.388 contains 7 formulas:


Section 7.389 contains a single formula.


Section 7.391 contains 11 formulas:


Section 7.392 contains 4 formulas:


Section 7.393 contains 2 formulas:


Section 7.411 contains 5 formulas:


Section 7.412 contains 2 formulas:


Section 7.413 contains a single formula.


Section 7.414 contains 13 formulas:


Section 7.415 contains a single formula.


Section 7.416 contains a single formula.


Section 7.417 contains 4 formulas:


Section 7.418 contains 6 formulas:


Section 7.419 contains a single formula.


Section 7.421 contains 6 formulas:


Section 7.422 contains 2 formulas:


Section 7.423 contains 2 formulas:


Section 7.511 contains a single formula.


Section 7.512 contains 12 formulas:


Section 7.513 contains a single formula.


Section 7.521 contains a single formula.


Section 7.522 contains 9 formulas:


Section 7.523 contains a single formula.


Section 7.524 contains 3 formulas:


Section 7.525 contains 2 formulas:


Section 7.526 contains 3 formulas:


Section 7.527 contains 4 formulas:


Section 7.531 contains 2 formulas:


Section 7.541 contains a single formula.


Section 7.542 contains 18 formulas:


Section 7.543 contains 2 formulas:


Section 7.544 contains a single formula.


Section 7.611 contains 7 formulas:


Section 7.612 contains 2 formulas:


Section 7.613 contains 6 formulas:


Section 7.621 contains 12 formulas:


Section 7.622 contains 3 formulas:


Section 7.623 contains 8 formulas:


Section 7.624 contains 6 formulas:


Section 7.625 contains 4 formulas:


Section 7.626 contains 2 formulas:


Section 7.627 contains 9 formulas:


Section 7.628 contains 3 formulas:


Section 7.629 contains 2 formulas:


Section 7.631 contains 3 formulas:


Section 7.632 contains a single formula.


Section 7.641 contains a single formula.


Section 7.642 contains a single formula.


Section 7.643 contains 4 formulas:


Section 7.644 contains 5 formulas:


Section 7.651 contains 2 formulas:


Section 7.652 contains a single formula.


Section 7.661 contains 6 formulas:


Section 7.662 contains 4 formulas:


Section 7.663 contains 7 formulas:


Section 7.664 contains 5 formulas:


Section 7.665 contains 2 formulas:


Section 7.666 contains a single formula.


Section 7.671 contains 2 formulas:


Section 7.672 contains 15 formulas:


Section 7.673 contains 2 formulas:


Section 7.674 contains 2 formulas:


Section 7.675 contains 4 formulas:


Section 7.681 contains 5 formulas:


Section 7.682 contains 5 formulas:


Section 7.683 contains a single formula.


Section 7.684 contains a single formula.


Section 7.691 contains a single formula.


Section 7.692 contains a single formula.


Section 7.693 contains 2 formulas:


Section 7.694 contains a single formula.


Section 7.711 contains 3 formulas:


Section 7.721 contains 2 formulas:


Section 7.722 contains 2 formulas:


Section 7.723 contains 3 formulas:


Section 7.724 contains a single formula.


Section 7.725 contains 6 formulas:


Section 7.726 contains a single formula.


Section 7.727 contains a single formula.


Section 7.728 contains a single formula.


Section 7.731 contains 2 formulas:


Section 7.741 contains 5 formulas:


Section 7.742 contains 4 formulas:


Section 7.743 contains a single formula.


Section 7.744 contains 2 formulas:


Section 7.751 contains 3 formulas:


Section 7.752 contains 18 formulas:


Section 7.753 contains 2 formulas:


Section 7.754 contains 4 formulas:


Section 7.755 contains 5 formulas:


Section 7.756 contains a single formula.


Section 7.761 contains 2 formulas:


Section 7.771 contains a single formula.


Section 7.772 contains 2 formulas:


Section 7.773 contains 2 formulas:


Section 7.774 contains a single formula.


Section 7.811 contains 5 formulas:


Section 7.812 contains 3 formulas:


Section 7.813 contains 2 formulas:


Section 7.814 contains 2 formulas:


Section 7.815 contains 2 formulas:


Section 7.821 contains 3 formulas:


Section 7.822 contains 3 formulas:


Section 7.823 contains 2 formulas:


Section 7.824 contains 2 formulas:


Section 7.831 contains a single formula.


Section 7.832 contains a single formula.



8.258: Integrals involving the complementary error function.

Section 8.258 contains 5 formulas:



8.259: Integrals involving the error function and an exponential function.

Section 8.259 contains 3 formulas: