# Cauchy sum

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##### 3: 2.10 Sums and Sequences
###### §2.10 Sums and Sequences
For an extension to integrals with Cauchy principal values see Elliott (1998). … and Cauchy’s theorem, we have … These problems can be brought within the scope of §2.4 by means of Cauchy’s integral formula … By allowing the contour in Cauchy’s formula to expand, we find that …
##### 4: 4.10 Integrals
The left-hand side of (4.10.7) is a Cauchy principal value (§1.4(v)). …
19.8.6 $E\left(k\right)=\frac{\pi}{2M\left(1,k^{\prime}\right)}\left(a_{0}^{2}-\sum_{n% =0}^{\infty}2^{n-1}c_{n}^{2}\right)=K\left(k\right)\left(a_{1}^{2}-\sum_{n=2}^% {\infty}2^{n-1}c_{n}^{2}\right),$ $-\infty, $a_{0}=1$, $g_{0}=k^{\prime}$,
19.8.7 $\Pi\left(\alpha^{2},k\right)=\frac{\pi}{4M\left(1,k^{\prime}\right)}\left(2+% \frac{\alpha^{2}}{1-\alpha^{2}}\sum_{n=0}^{\infty}Q_{n}\right),$ $-\infty, $-\infty<\alpha^{2}<1$,
If $\alpha^{2}>1$, then the Cauchy principal value is
19.8.9 $\Pi\left(\alpha^{2},k\right)=\frac{\pi}{4M\left(1,k^{\prime}\right)}\frac{k^{2% }}{k^{2}-\alpha^{2}}\sum_{n=0}^{\infty}Q_{n},$ $-\infty, $1<\alpha^{2}<\infty$,
##### 6: 2.3 Integrals of a Real Variable
is finite and bounded for $n=0,1,2,\dots$, then the $n$th error term (that is, the difference between the integral and $n$th partial sum in (2.3.2)) is bounded in absolute value by $|q^{(n)}(0)/(x^{n}(x-\sigma_{n}))|$ when $x$ exceeds both $0$ and $\sigma_{n}$. …
2.3.4 $\int_{a}^{b}e^{ixt}q(t)\mathrm{d}t\sim e^{iax}\sum_{s=0}^{\infty}q^{(s)}(a)% \left(\frac{i}{x}\right)^{s+1}-e^{ibx}\sum_{s=0}^{\infty}q^{(s)}(b)\left(\frac% {i}{x}\right)^{s+1},$ $x\to+\infty$.
• (b)

As $t\to a+$

2.3.14
$p(t)\sim p(a)+\sum_{s=0}^{\infty}p_{s}(t-a)^{s+\mu},$
$q(t)\sim\sum_{s=0}^{\infty}q_{s}(t-a)^{s+\lambda-1},$

and the expansion for $p(t)$ is differentiable. Again $\lambda$ and $\mu$ are positive constants. Also $p_{0}>0$ (consistent with (a)).

• But if (d) applies, then the second sum is absent. …
##### 7: 19.21 Connection Formulas
19.21.11 $6R_{G}\left(x,y,z\right)=3(x+y+z)R_{F}\left(x,y,z\right)-\sum x^{2}R_{D}\left(% y,z,x\right)=\sum x(y+z)R_{D}\left(y,z,x\right),$
The latter case allows evaluation of Cauchy principal values (see (19.20.14)). …
19.22.9 $\frac{4}{\pi}R_{G}\left(0,a_{0}^{2},g_{0}^{2}\right)=\frac{1}{M\left(a_{0},g_{% 0}\right)}\left(a_{0}^{2}-\sum_{n=0}^{\infty}2^{n-1}c_{n}^{2}\right)=\frac{1}{% M\left(a_{0},g_{0}\right)}\left(a_{1}^{2}-\sum_{n=2}^{\infty}2^{n-1}c_{n}^{2}% \right),$
19.22.10 $R_{D}\left(0,g_{0}^{2},a_{0}^{2}\right)=\frac{3\pi}{4M\left(a_{0},g_{0}\right)% a_{0}^{2}}\sum_{n=0}^{\infty}Q_{n},$
19.22.12 $R_{J}\left(0,g_{0}^{2},a_{0}^{2},p_{0}^{2}\right)=\frac{3\pi}{4M\left(a_{0},g_% {0}\right)p_{0}^{2}}\sum_{n=0}^{\infty}Q_{n},$
If the last variable of $R_{J}$ is negative, then the Cauchy principal value is
19.22.14 $R_{J}\left(0,g_{0}^{2},a_{0}^{2},-q_{0}^{2}\right)=\frac{-3\pi}{4M\left(a_{0},% g_{0}\right)(q_{0}^{2}+a_{0}^{2})}\*\left(2+\frac{a_{0}^{2}-g_{0}^{2}}{q_{0}^{% 2}+g_{0}^{2}}\sum_{n=0}^{\infty}Q_{n}\right),$
##### 9: 1.3 Determinants
1.3.9 $\det[a_{jk}]^{2}\leq\left(\sum^{n}_{k=1}a^{2}_{1k}\right)\left(\sum^{n}_{k=1}a% ^{2}_{2k}\right)\dots\left(\sum^{n}_{k=1}a^{2}_{nk}\right).$
for every distinct pair of $j,k$, or when one of the factors $\sum^{n}_{k=1}a^{2}_{jk}$ vanishes. …
###### Cauchy Determinant
1.3.19 $\sum^{\infty}_{j,k=-\infty}|a_{j,k}-\delta_{j,k}|$
3.5.5 $\int_{-\infty}^{\infty}f(t)\mathrm{d}t=h\sum_{k=-\infty}^{\infty}f(kh)+E_{h}(f),$
3.5.12 $G_{0}(\tfrac{1}{2}h)=\tfrac{1}{2}G_{0}(h)+\tfrac{1}{2}h\sum_{k=0}^{n-1}f\left(% x_{0}+(k+\tfrac{1}{2})h\right),$
3.5.46 $\mathcal{H}\mskip-3.0mu f\mskip 3.0mu \left(x\right)=\frac{1}{\pi}\pvint_{-% \infty}^{\infty}\frac{f(t)}{t-x}\mathrm{d}t,$ $x\in\mathbb{R}$,