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21: 19.21 Connection Formulas

§19.21 Connection Formulas

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§19.21(i) Complete Integrals

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§19.21(ii) Incomplete Integrals

… ►where both summations extend over the three cyclic permutations of $x,y,z$. … ►

§19.21(iii) Change of Parameter of $R_J$

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22: 29.14 Orthogonality

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29.14.2 $$⟨g,h⟩={\int }_0^K{\int }_0^{K^{\prime }}w(s,t)g(s,t)h(s,t)dtds,$$

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Symbols:

$K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $dx$: differential of $x$ , $\int$: integral, $k$: real parameter and $w(s,t)$: function

Referenced by:

§29.14

Permalink:

http://dlmf.nist.gov/29.14.E2

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Annotations for §29.14 and Ch.29

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29.14.3 $$w(s,t)={\mathrm{sn}}^2(K+\mathrm{i}t,k)-{\mathrm{sn}}^2(s,k).$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $k$: real parameter and $w(s,t)$: function

Referenced by:

§29.14

Permalink:

http://dlmf.nist.gov/29.14.E3

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Annotations for §29.14 and Ch.29

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29.14.4 $${\mathrm{𝑠𝐸}}_{2n+1}^m(s,k^2){\mathrm{𝑠𝐸}}_{2n+1}^m(K+\mathrm{i}t,k^2),$$

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Symbols:

${\mathrm{𝑠𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer, $n$: nonnegative integer and $k$: real parameter

Referenced by:

§29.14

Permalink:

http://dlmf.nist.gov/29.14.E4

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Annotations for §29.14 and Ch.29

… ►In each system $n$ ranges over all nonnegative integers and $m=0,1,\mathrm{\dots },n$. … ►

29.14.11 $$⟨g,h⟩={\int }_0^{4K}{\int }_0^{2K^{\prime }}w(s,t)g(s,t)h(s,t)dtds,$$

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Symbols:

$K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $dx$: differential of $x$ , $\int$: integral, $k$: real parameter and $w(s,t)$: function

Permalink:

http://dlmf.nist.gov/29.14.E11

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Annotations for §29.14 and Ch.29

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23: 19.38 Approximations

§19.38 Approximations

… ►Approximations of the same type for $K\left(k\right)$ and $E\left(k\right)$ for are given in Cody (1965a) with maximum absolute errors ranging from 4×10⁻⁵ to 4×10⁻¹⁸. … ►Approximations for Legendre’s complete or incomplete integrals of all three kinds, derived by Padé approximation of the square root in the integrand, are given in Luke (1968, 1970). They are valid over parts of the complex $k$ and $\varphi$ planes. …

24: 30.15 Signal Analysis

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§30.15(ii) Integral Equation

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30.15.7 $${\int }_{-\tau }^{\tau }{\varphi }_k(t){\varphi }_n(t)dt={\mathrm{\Lambda }}_n{\delta }_{k,n},$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\int$: integral, $n\ge m$: integer degree, $\tau >0$: parameter, ${\varphi }_n(t)$: function and $\mathrm{\Lambda }$

Permalink:

http://dlmf.nist.gov/30.15.E7

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Annotations for §30.15(iv), §30.15 and Ch.30

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30.15.8 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\varphi }_k(t){\varphi }_n(t)dt={\delta }_{k,n}.$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\int$: integral, $n\ge m$: integer degree and ${\varphi }_n(t)$: function

Permalink:

http://dlmf.nist.gov/30.15.E8

Encodings:

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Annotations for §30.15(iv), §30.15 and Ch.30

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30.15.9 $$\beta =\frac{1}{2\pi }{\int }_{-\sigma }^{\sigma }{\left|{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-\mathrm{i}t\omega }f(t)dt\right|}^{2}d\omega$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sigma >0$: parameter and $f$: function

Permalink:

http://dlmf.nist.gov/30.15.E9

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Annotations for §30.15(v), §30.15 and Ch.30

►taken over all $f\in L^2(-\mathrm{\infty },\mathrm{\infty })$ subject to …

25: 28.30 Expansions in Series of Eigenfunctions

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28.30.2 $$\frac{1}{2\pi }{\int }_0^{2\pi }{w}_m(x)w_n(x)dx={\delta }_{m,n}.$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $m$: integer, $n$: integer, $x$: real variable and $w(z)$: Mathieu’s equation solution

Permalink:

http://dlmf.nist.gov/28.30.E2

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Annotations for §28.30(i), §28.30 and Ch.28

►Then every continuous $2\pi$-periodic function $f(x)$ whose second derivative is square-integrable over the interval $[0,2\pi ]$ can be expanded in a uniformly and absolutely convergent series … ►

28.30.4 $$f_m=\frac{1}{2\pi }{\int }_0^{2\pi }f(x)w_m(x)dx.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $m$: integer, $x$: real variable and $w(z)$: Mathieu’s equation solution

Permalink:

http://dlmf.nist.gov/28.30.E4

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Annotations for §28.30(i), §28.30 and Ch.28

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26: 19.19 Taylor and Related Series

§19.19 Taylor and Related Series

… ►where the summation extends over all nonnegative integers $m_1,\mathrm{\dots },m_n$ whose sum is $N$. The following two multivariate hypergeometric series apply to each of the integrals (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23): … ►where $M={\sum }_{j=1}^n{m}_j$ and the summation extends over all nonnegative integers $m_1,\mathrm{\dots },m_n$ such that ${\sum }_{j=1}^{n}j{m}_j=N$. … ►

27: 19.23 Integral Representations

§19.23 Integral Representations

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19.23.1 $$R_F(0,y,z)={\int }_0^{\pi /2}{(y{\cos }^2\theta +z{\sin }^2\theta )}^{-1/2}d\theta ,$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral and $\sin z$: sine function

Referenced by:

§19.23, §19.23

Permalink:

http://dlmf.nist.gov/19.23.E1

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Annotations for §19.23 and Ch.19

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19.23.4 $$R_F(0,y,z)=\frac{2}{\pi }{\int }_0^{\pi /2}{R}_C(y,z{\cos }^2\theta )d\theta =\frac{2}{\pi }{\int }_0^{\mathrm{\infty }}{R}_C(y{\cosh }^{2}t,z)dt.$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function and $\int$: integral

Referenced by:

§19.23

Permalink:

http://dlmf.nist.gov/19.23.E4

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Annotations for §19.23 and Ch.19

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19.23.5 $$R_F(x,y,z)=\frac{2}{\pi }{\int }_0^{\pi /2}{R}_C(x,y{\cos }^2\theta +z{\sin }^2\theta )d\theta ,$$ $\mathrm{\Re }y>0$, $\mathrm{\Re }z>0$,

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part and $\sin z$: sine function

Referenced by:

§19.2(iv), §19.23

Permalink:

http://dlmf.nist.gov/19.23.E5

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Annotations for §19.23 and Ch.19

… ►In (19.23.8)–(19.23.10) one or more of the variables may be 0 if the integral converges. …

28: 2.10 Sums and Sequences

… ►Assume that $a,m$, and $n$ are integers such that $n>a$, $m>0$, and $f^{(2m)}(x)$ is absolutely integrable over $[a,n]$. … ►

The first infinite integral in (2.10.2) converges.

… ►For an extension to integrals with Cauchy principal values see Elliott (1998). … ►These problems can be brought within the scope of §2.4 by means of Cauchy’s integral formula … ►In these circumstances the integrals in (2.10.28) are integrable by parts $m$ times, yielding …

29: 2.5 Mellin Transform Methods

… ►when this integral converges. … ►The sum in (2.5.6) is taken over all poles of $x^{-z}ℳf\left(1-z\right)ℳh\left(z\right)$ in the strip , and it provides the asymptotic expansion of $I(x)$ for small values of $x$. … ►To ensure that the integral (2.5.3) converges we assume that … ►where … ►To verify (2.5.48) we may use …

30: 1.8 Fourier Series

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1.8.4 $$c_n=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }f(x){\mathrm{e}}^{-\mathrm{i}nx}dx.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral and $n$: nonnegative integer

Referenced by:

§1.17(iii), §1.18(v), §1.8(i)

Permalink:

http://dlmf.nist.gov/1.8.E4

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Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

… ►(1.8.10) continues to apply if either $a$ or $b$ or both are infinite and/or $f(x)$ has finitely many singularities in $(a,b)$, provided that the integral converges uniformly (§1.5(iv)) at $a,b$, and the singularities for all sufficiently large $\lambda$. … ►Suppose that $f(x)$ is twice continuously differentiable and $f(x)$ and $\left|f^{\prime \prime }(x)\right|$ are integrable over $(-\mathrm{\infty },\mathrm{\infty })$. …It follows from definition (1.14.1) that the integral in (1.8.14) is equal to $\sqrt{2\pi }ℱ\left(f\right)\left(-2\pi n\right)$. … ►

1.8.15 $$\frac{1}{2}f(0)+\sum _{n=1}^{\mathrm{\infty }}f(n)={\int }_0^{\mathrm{\infty }}f(x)dx+2\sum _{n=1}^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}f(x)\cos \left(2\pi nx\right)dx.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/1.8.E15

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Annotations for §1.8(iv), §1.8 and Ch.1

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