Cauchy theorem

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2: 2.10 Sums and Sequences
and Cauchy’s theorem, we have …
5: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
An inner product space $V$ is called a Hilbert space if every Cauchy sequence $\{v_{n}\}$ in $V$ (i. …
1.18.54 $\lim_{\epsilon\to 0{+}}\int_{X}\frac{f(y)}{x\pm\mathrm{i}\epsilon-y}\,\mathrm{% d}y=P\pvint_{X}\frac{f(y)}{x-y}\,\mathrm{d}y\mp\mathrm{i}\pi f(x).$
1.18.57 $\lim_{R\to\infty}\int_{0}^{\infty}f(y)\left(\int_{0}^{R}\sqrt{xt}J_{\nu}\left(% xt\right)\sqrt{yt}J_{\nu}\left(yt\right)\,\mathrm{d}t\right)\,\mathrm{d}y=% \tfrac{1}{2}\left(f(x+)+f(x-)\right),$ $x>0$, $\nu>-1$,
Surprisingly, if $q(x)<0$ on any interval on the real line, even if positive elsewhere, as long as $\int_{X}q(x)\,\mathrm{d}x\leq 0$, see Simon (1976, Theorem 2.5), then there will be at least one eigenfunction with a negative eigenvalue, with corresponding $L^{2}\left(X\right)$ eigenfunction. …
6: 18.17 Integrals
18.17.42 $\pvint_{-1}^{1}T_{n}\left(y\right)\frac{(1-y^{2})^{-\frac{1}{2}}}{y-x}\,% \mathrm{d}y=\pi U_{n-1}\left(x\right),$
18.17.43 $\pvint_{-1}^{1}U_{n-1}\left(y\right)\frac{(1-y^{2})^{\frac{1}{2}}}{y-x}\,% \mathrm{d}y=-\pi T_{n}\left(x\right).$
These integrals are Cauchy principal values (§1.4(v)). …
7: 19.36 Methods of Computation
Numerical differences between the variables of a symmetric integral can be reduced in magnitude by successive factors of 4 by repeated applications of the duplication theorem, as shown by (19.26.18). … All cases of $R_{F}$, $R_{C}$, $R_{J}$, and $R_{D}$ are computed by essentially the same procedure (after transforming Cauchy principal values by means of (19.20.14) and (19.2.20)). … When the values of complete integrals are known, addition theorems with $\psi=\pi/2$19.11(ii)) ease the computation of functions such as $F\left(\phi,k\right)$ when $\frac{1}{2}\pi-\phi$ is small and positive. …These special theorems are also useful for checking computer codes. …