Complex Analysis Examples
Some examples taken from Applied Complex Analysis, by Henrici
p. 421
\int^1_\kappa
\left[\bigl(1-w^2\bigr)\bigl(\kappa^2-w^2\bigr)\right]^{-1/2} dw
= \frac{4}{\left(1+\sqrt{\kappa}\,\right)^2} K
\left(\left(\frac{1-\sqrt{\kappa}}{1+\sqrt{\kappa}}\right)^{\!\!2}\right)
p. 425
\mathop{\rm grd} \phi(z) =
\left(a+\frac{2d}{\pi}\right) v_\infty\, \overline{f'(z)} =
v_\infty \left[ \pi a + \frac{2d}{\pi a + 2dw^{-1/2}(w-1)^{1/2}} \right]^-
p. 455
-\sum^n_{m=1}
\left(\,\sum^\infty_{k=1} \frac{ h^{k-1} }{\left(w_m-z_0\right)^2}
\right) = \sum^\infty_{k=1} s_k\, h^{k-1}