Cauchy theorem

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1: 1.9 Calculus of a Complex Variable

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Cauchy’s Theorem

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2: 2.10 Sums and Sequences

… ►and Cauchy’s theorem, we have …

3: 1.4 Calculus of One Variable

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Mean Value Theorem

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Cauchy Principal Values

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Fundamental Theorem of Calculus

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First Mean Value Theorem

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Second Mean Value Theorem

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4: 17.5 ${}_0{}^{}\varphi _0^{},{}_1{}^{}\varphi _0^{},{}_1{}^{}\varphi _1^{}$ Functions

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$q$-Binomial Theorem

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Cauchy’s Sum

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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 $$\underset{ϵ\to 0+}{lim}{\int }_X\frac{f(y)}{x\pm \mathrm{i}ϵ-y}dy=P{⨍}_X\frac{f(y)}{x-y}dy\mp \mathrm{i}\pi f(x).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and ${⨍}_a^b$: Cauchy principal value

Permalink:

http://dlmf.nist.gov/1.18.E54

Encodings:

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See also:

Annotations for §1.18(vi), §1.18(vi), §1.18 and Ch.1

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1.18.57 $$\underset{R\to \mathrm{\infty }}{lim}{\int }_0^{\mathrm{\infty }}f(y)\left({\int }_0^R\sqrt{xt}{J}_{\nu }\left(xt\right)\sqrt{yt}{J}_{\nu }\left(yt\right)dt\right)dy=\frac{1}{2}\left(f(x+)+f(x-)\right),$$ $x>0$, $\nu >-1$,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $dx$: differential of $x$ and $\int$: integral

Proved:

Li and Wong (2008, §3, Theorem 1)(proved)

Referenced by:

§1.18(vi), §1.18(vi)

Permalink:

http://dlmf.nist.gov/1.18.E57

Encodings:

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See also:

Annotations for §1.18(vi), §1.18(vi), §1.18 and Ch.1

… ►Surprisingly, if on any interval on the real line, even if positive elsewhere, as long as ${\int }_{X}q(x)dx\le 0$, 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

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18.17.42 $${⨍}_{-1}^1{T}_n\left(y\right)\frac{{(1-y^2)}^{-\frac{1}{2}}}{y-x}dy=\pi U_{n-1}\left(x\right),$$

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Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $U_n\left(x\right)$: Chebyshev polynomial of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , ${⨍}_a^b$: Cauchy principal value, $y$: real variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.13.3

Referenced by:

§18.17(viii)

Permalink:

http://dlmf.nist.gov/18.17.E42

Encodings:

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See also:

Annotations for §18.17(viii), §18.17(viii), §18.17 and Ch.18

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18.17.43 $${⨍}_{-1}^1{U}_{n-1}\left(y\right)\frac{{(1-y^2)}^{\frac{1}{2}}}{y-x}dy=-\pi T_n\left(x\right).$$

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Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $U_n\left(x\right)$: Chebyshev polynomial of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , ${⨍}_a^b$: Cauchy principal value, $y$: real variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.13.4

Referenced by:

§18.17(viii)

Permalink:

http://dlmf.nist.gov/18.17.E43

Encodings:

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See also:

Annotations for §18.17(viii), §18.17(viii), §18.17 and Ch.18

►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 $F(\varphi ,k)$ when $\frac{1}{2}\pi -\varphi$ is small and positive. …These special theorems are also useful for checking computer codes. …