Cauchy sum

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1: 17.5 ${}_0{}^{}\varphi _0^{},{}_1{}^{}\varphi _0^{},{}_1{}^{}\varphi _1^{}$ Functions

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Cauchy’s Sum

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2: 1.7 Inequalities

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Cauchy–Schwarz Inequality

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Cauchy–Schwarz Inequality

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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

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4.10.7 $${⨍}_0^x\frac{dt}{\ln t}=\mathrm{li}\left(x\right),$$ $x>1$.

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Symbols:

$dx$: differential of $x$ , $\mathrm{li}\left(x\right)$: logarithmic integral, $\ln z$: principal branch of logarithm function, ${⨍}_a^b$: Cauchy principal value and $x$: real variable

A&S Ref:

4.1.57

Referenced by:

§4.10(i)

Permalink:

http://dlmf.nist.gov/4.10.E7

Encodings:

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See also:

Annotations for §4.10(i), §4.10 and Ch.4

►The left-hand side of (4.10.7) is a Cauchy principal value (§1.4(v)). …

5: 18.40 Methods of Computation

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18.40.5 $$F_N(z)=\frac{1}{{\mu }_0}\sum _{n=1}^N\frac{w_n}{z-x_n}.$$

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Symbols:

$N$: positive integer, $w_x$: weights, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.40.E5

Encodings:

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See also:

Annotations for §18.40(ii), §18.40(ii), §18.40 and Ch.18

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18.40.6 $$\underset{\epsilon \to 0+}{lim}{\int }_a^b\frac{w(x)dx}{x^{\prime }+\mathrm{i}\epsilon -x}dx={⨍}_a^b\frac{w(x)dx}{x^{\prime }-x}-\mathrm{i}\pi w(x^{\prime }),$$

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Symbols:

$dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, ${⨍}_a^b$: Cauchy principal value, $w(x)$: weight function and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.40.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.40(ii), §18.40(ii), §18.40 and Ch.18

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18.40.7 $${\mu }_N(x)=\sum _{n=1}^N{w}_{n}H\left(x-x_n\right),$$ $x\in (a,b)$,

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Symbols:

$H\left(x\right)$: Heaviside function, $\in$: element of, $(a,b)$: open interval, $N$: positive integer, $w_x$: weights, $n$: nonnegative integer and $x$: real variable

Source:

Shohat and Tamarkin (1970, pp. 42-43)

Referenced by:

§18.40(ii)

Permalink:

http://dlmf.nist.gov/18.40.E7

Encodings:

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See also:

Annotations for §18.40(ii), §18.40(ii), §18.40 and Ch.18

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6: 1.4 Calculus of One Variable

… ►where the sum is over all nonnegative integers $m_1,m_2,\mathrm{\dots },m_n$ that satisfy $m_1+2m_2+\mathrm{\cdots }+n{m}_n=n$, and $k=m_1+m_2+\mathrm{\cdots }+m_n$. … ►Definite integrals over the Stieltjes measure $d\alpha (x)$ could represent a sum, an integral, or a combination of the two. Let $d\alpha (x)=w(x)dx+{\sum }_{n=1}^N{w}_n\delta \left(x-x_n\right)dx$, $x_n\in (a,b)$, $n=1,\mathrm{\dots }N$. … ►

Cauchy Principal Values

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1.4.24 $${⨍}_a^{b}f(x)dx=𝑃{\int }_a^{b}f(x)dx=\underset{ϵ\to 0+}{lim}\left({\int }_a^{c-ϵ}f(x)dx+{\int }_{c+ϵ}^{b}f(x)dx\right),$$

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Defines:

${⨍}_a^b$: Cauchy principal value

Symbols:

$dx$: differential of $x$ and $\int$: integral

Referenced by:

§1.18(vi)

Permalink:

http://dlmf.nist.gov/1.4.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

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7: 18.17 Integrals

… ►provided that $\mathrm{\ell }+m+n$ is even and the sum of any two of $\mathrm{\ell },m,n$ is not less than the third; otherwise the integral is zero. … ►

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)). … ►provided that $\mathrm{\ell }+m+n$ is even and the sum of any two of $\mathrm{\ell },m,n$ is not less than the third; otherwise the integral is zero. …

8: 19.8 Quadratic Transformations

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19.8.6 $$E\left(k\right)=\frac{\pi }{2M(1,k^{\prime })}\left(a_0^2-\sum _{n=0}^{\mathrm{\infty }}{2}^{n-1}{c}_n^2\right)=K\left(k\right)\left(a_1^2-\sum _{n=2}^{\mathrm{\infty }}{2}^{n-1}{c}_n^2\right),$$ , $a_0=1$, $g_0=k^{\prime }$,

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Symbols:

$M(a,g)$: arithmetic-geometric mean, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $n$: nonnegative integer, $k$: real or complex modulus, $k^{\prime }$: complementary modulus, $a_n$: iterate and $g_n$: iterate

Permalink:

http://dlmf.nist.gov/19.8.E6

Encodings:

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See also:

Annotations for §19.8(i), §19.8 and Ch.19

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19.8.7 $$\mathrm{\Pi }({\alpha }^2,k)=\frac{\pi }{4M(1,k^{\prime })}\left(2+\frac{{\alpha }^2}{1-{\alpha }^2}\sum _{n=0}^{\mathrm{\infty }}{Q}_n\right),$$ , ,

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Symbols:

$M(a,g)$: arithmetic-geometric mean, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Pi }({\alpha }^2,k)$: Legendre’s complete elliptic integral of the third kind, $n$: nonnegative integer, $k$: real or complex modulus, $k^{\prime }$: complementary modulus, ${\alpha }^2$: real or complex parameter and $Q_n$

Referenced by:

§19.8(i)

Permalink:

http://dlmf.nist.gov/19.8.E7

Encodings:

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See also:

Annotations for §19.8(i), §19.8 and Ch.19

… ►If ${\alpha }^2>1$, then the Cauchy principal value is ►

19.8.9 $$\mathrm{\Pi }({\alpha }^2,k)=\frac{\pi }{4M(1,k^{\prime })}\frac{k^2}{k^2-{\alpha }^2}\sum _{n=0}^{\mathrm{\infty }}{Q}_n,$$ , ,

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Symbols:

$M(a,g)$: arithmetic-geometric mean, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Pi }({\alpha }^2,k)$: Legendre’s complete elliptic integral of the third kind, $n$: nonnegative integer, $k$: real or complex modulus, $k^{\prime }$: complementary modulus, ${\alpha }^2$: real or complex parameter and $Q_n$

Referenced by:

§19.8(i)

Permalink:

http://dlmf.nist.gov/19.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.8(i), §19.8 and Ch.19

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9: 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. … ►where the infinite sum means convergence in norm, … ►The sum of the kinetic and potential energies give the quantum Hamiltonian, or energy operator; often also referred to as a Schrödinger operator. … ►Should an eigenvalue correspond to more than a single linearly independent eigenfunction, namely a multiplicity greater than one, all such eigenfunctions will always be implied as being part of any sums or integrals over the spectrum. … ►Spectral expansions of $T$, and of functions $F(T)$ of $T$, these being expansions of $T$ and $F(T)$ in terms of the eigenvalues and eigenfunctions summed over the spectrum, then follow: …

10: 2.3 Integrals of a Real Variable

… ►is finite and bounded for $n=0,1,2,\mathrm{\dots }$, 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 $${\int }_a^b{\mathrm{e}}^{\mathrm{i}xt}q(t)dt\sim {\mathrm{e}}^{\mathrm{i}ax}\sum _{s=0}^{\mathrm{\infty }}{q}^{(s)}(a){\left(\frac{\mathrm{i}}{x}\right)}^{s+1}-{\mathrm{e}}^{\mathrm{i}bx}\sum _{s=0}^{\mathrm{\infty }}{q}^{(s)}(b){\left(\frac{\mathrm{i}}{x}\right)}^{s+1},$$ $x\to +\mathrm{\infty }$.

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Symbols:

$\sim$: Poincaré asymptotic expansion, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $q(t)$: infinitely differentiable function, $a$: left endpoint and $b$: right endpoint

Permalink:

http://dlmf.nist.gov/2.3.E4

Encodings:

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See also:

Annotations for §2.3(i), §2.3 and Ch.2

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As $t\to a+$

2.3.14

$$p(t)\sim p(a)+\sum _{s=0}^{\mathrm{\infty }}{p}_s{(t-a)}^{s+\mu },$$

$$q(t)\sim \sum _{s=0}^{\mathrm{\infty }}{q}_s{(t-a)}^{s+\lambda -1},$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $a$: left endpoint, $p(t)$: real function, $p_s$: coefficients, $q(t)$: function, $q_s$: coefficients, $\lambda$: positive constant and $\mu$: positive constant

Referenced by:

item (b)

Permalink:

http://dlmf.nist.gov/2.3.E14

Encodings:

pMML, pMML, png, png, TeX, TeX

See also:

Annotations for §2.3(iii), §2.3 and Ch.2

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. … ►

2.3.31 $$f(\alpha ,w)=\sum _{s=0}^{\mathrm{\infty }}{\varphi }_s(\alpha ){(w-a)}^s,$$

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Symbols:

$w$: change of variable, $a(\alpha )$ , $f(\alpha ,w)$: function and ${\varphi }_s(\alpha )$: coefficients

Permalink:

http://dlmf.nist.gov/2.3.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §2.3(v), §2.3 and Ch.2

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