change of parameter

SearchAdvancedHelp

(0.003 seconds)

21—30 of 54 matching pages

21: 28.14 Fourier Series

… ►

28.14.4 $$q{c}_{2m+2}-\left(a-{(\nu +2m)}^2\right)c_{2m}+q{c}_{2m-2}=0,$$ $a={\lambda }_{\nu }\left(q\right),c_{2m}=c_{2m}^{\nu }(q)$,

ⓘ

Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $m$: integer, $q=h^2$: parameter, $\nu$: complex parameter, $a$: parameter and $c_{2m}(q)$: coefficients

Referenced by:

item (d), item (d)

Permalink:

http://dlmf.nist.gov/28.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.14 and Ch.28

… ►

28.14.5 $$\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{\left(c_{2m}^{\nu }(q)\right)}^2=1;$$

ⓘ

Symbols:

$m$: integer, $q=h^2$: parameter, $\nu$: complex parameter and $c_{2m}(q)$: coefficients

Referenced by:

item (d)

Permalink:

http://dlmf.nist.gov/28.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.14 and Ch.28

… ►For changes of sign of $\nu$, $q$, and $m$, ►

28.14.7 $$c_{-2m}^{-\nu }(q)=c_{2m}^{\nu }(q),$$

ⓘ

Symbols:

$m$: integer, $q=h^2$: parameter, $\nu$: complex parameter and $c_{2m}(q)$: coefficients

Permalink:

http://dlmf.nist.gov/28.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.14 and Ch.28

►

28.14.8 $$c_{2m}^{\nu }(-q)={(-1)}^m{c}_{2m}^{\nu }(q).$$

ⓘ

Symbols:

$m$: integer, $q=h^2$: parameter, $\nu$: complex parameter and $c_{2m}(q)$: coefficients

Permalink:

http://dlmf.nist.gov/28.14.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §28.14 and Ch.28

…

22: 28.12 Definitions and Basic Properties

… ►For change of signs of $\nu$ and $q$, … ►For changes of sign of $\nu$, $q$, and $z$, ►

28.12.8 $${\mathrm{me}}_{-\nu }(z,q)={\mathrm{me}}_{\nu }(-z,q),$$

ⓘ

Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $q=h^2$: parameter, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.12.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §28.12(ii), §28.12 and Ch.28

… ►For change of signs of $\nu$ and $z$, ►

28.12.14 $${\mathrm{ce}}_{\nu }(z,q)={\mathrm{ce}}_{\nu }(-z,q)={\mathrm{ce}}_{-\nu }(z,q),$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $q=h^2$: parameter, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.12.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §28.12(iii), §28.12 and Ch.28

…

23: 2.5 Mellin Transform Methods

… ►

2.5.41 $$I_1(x)=ℳh_1\left(1\right)x^{-1}+\frac{1}{2\pi \mathrm{i}}{\int }_{\rho -\mathrm{i}\mathrm{\infty }}^{\rho +\mathrm{i}\mathrm{\infty }}{x}^{-z}\mathrm{\Gamma }\left(1-z\right)ℳh_1\left(z\right)dz,$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $ℳ\left(f\right)\left(s\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $h(x)$: locally integrable function, $I_j(x)$: integral and $\rho$: parameter

Keywords:

Mellin transform

Referenced by:

§2.5(iii)

Permalink:

http://dlmf.nist.gov/2.5.E41

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Mellin transform was changed to $ℳf\left(z\right)$ from $ℳ(f;z)$.

See also:

Annotations for §2.5(iii), §2.5 and Ch.2

… ►

2.5.42 $$I_2(x)=\sum _{\mathrm{\Re }{\beta }_0\le \mathrm{\Re }z\le 1}res\left[-x^{-z}\mathrm{\Gamma }\left(1-z\right)ℳh_2\left(z\right)\right]+\frac{1}{2\pi \mathrm{i}}{\int }_{\rho -\mathrm{i}\mathrm{\infty }}^{\rho +\mathrm{i}\mathrm{\infty }}{x}^{-z}\mathrm{\Gamma }\left(1-z\right)ℳh_2\left(z\right)dz.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $ℳ\left(f\right)\left(s\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $res$: residue, $h(x)$: locally integrable function, $I_j(x)$: integral and $\rho$: parameter

Keywords:

Mellin transform

Referenced by:

§2.5(iii)

Permalink:

http://dlmf.nist.gov/2.5.E42

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Mellin transform was changed to $ℳf\left(z\right)$ from $ℳ(f;z)$.

See also:

Annotations for §2.5(iii), §2.5 and Ch.2

…

24: 1.14 Integral Transforms

… ►

1.14.33 $$\underset{t\to \pm \mathrm{\infty }}{lim}ℳf\left(\sigma +\mathrm{i}t\right)=0.$$

ⓘ

Symbols:

$ℳ\left(f\right)\left(s\right)$: Mellin transform, $\mathrm{i}$: imaginary unit and $\sigma \in (a,b)$: parameter

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/1.14.E33

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Mellin transform was changed to $ℳf\left(s\right)$ from $ℳ(f;s)$.

See also:

Annotations for §1.14(iv), §1.14 and Ch.1

… ►

1.14.34 $$\frac{1}{2}(f(u+)+f(u-))=\frac{1}{2\pi \mathrm{i}}\underset{T\to \mathrm{\infty }}{lim}{\int }_{\sigma -\mathrm{i}T}^{\sigma +\mathrm{i}T}{u}^{-s}ℳf\left(s\right)ds.$$

ⓘ

Symbols:

$ℳ\left(f\right)\left(s\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $\sigma \in (a,b)$: parameter

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/1.14.E34

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Mellin transform was changed to $ℳf\left(s\right)$ from $ℳ(f;s)$.

See also:

Annotations for §1.14(iv), §1.14(iv), §1.14 and Ch.1

… ►

1.14.35 $$f(x)=\frac{1}{2\pi \mathrm{i}}{\int }_{\sigma -\mathrm{i}\mathrm{\infty }}^{\sigma +\mathrm{i}\mathrm{\infty }}{x}^{-s}ℳf\left(s\right)ds.$$

ⓘ

Symbols:

$ℳ\left(f\right)\left(s\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $\sigma \in (a,b)$: parameter

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/1.14.E35

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Mellin transform was changed to $ℳf\left(s\right)$ from $ℳ(f;s)$.

See also:

Annotations for §1.14(iv), §1.14(iv), §1.14 and Ch.1

… ►

1.14.36 $${\int }_0^{\mathrm{\infty }}f(x)g(yx)dx=\frac{1}{2\pi \mathrm{i}}{\int }_{\sigma -\mathrm{i}\mathrm{\infty }}^{\sigma +\mathrm{i}\mathrm{\infty }}{y}^{-s}ℳf\left(1-s\right)ℳg\left(s\right)ds,$$

ⓘ

Symbols:

$ℳ\left(f\right)\left(s\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $\sigma \in (a,b)$: parameter

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/1.14.E36

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Mellin transform was changed to $ℳf\left(s\right)$ from $ℳ(f;s)$.

See also:

Annotations for §1.14(iv), §1.14(iv), §1.14 and Ch.1

… ►

1.14.49 $$\underset{t\to 0+}{lim}\frac{𝒮f\left(-\sigma -\mathrm{i}t\right)-𝒮f\left(-\sigma +\mathrm{i}t\right)}{2\pi \mathrm{i}}=\frac{1}{2}(f(\sigma +)+f(\sigma -)),$$

ⓘ

Symbols:

$𝒮\left(f\right)\left(s\right)$: Stieltjes transform, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit and $\sigma \in (a,b)$: parameter

Permalink:

http://dlmf.nist.gov/1.14.E49

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Stieltjes transform was changed to $𝒮f\left(s\right)$ from $𝒮(f;s)$.

See also:

Annotations for §1.14(vi), §1.14(vi), §1.14 and Ch.1

…

25: 19.2 Definitions

… ►

19.2.11 $$\mathrm{cel}(k_c,p,a,b)={\int }_0^{\pi /2}\frac{a{\cos }^2\theta +b{\sin }^2\theta }{{\cos }^2\theta +p{\sin }^2\theta }\frac{d\theta }{\sqrt{{\cos }^2\theta +k_c^2{\sin }^2\theta }},$$

ⓘ

Defines:

$\mathrm{cel}(k_c,p,a,b)$: Bulirsch’s complete elliptic integral

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $a$: real parameter, $b$: real parameter, $p$: real parameter not equal to zero and $k_c$: real or complex variable not equal to zero

Referenced by:

§19.2(iii)

Permalink:

http://dlmf.nist.gov/19.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §19.2(iii), §19.2 and Ch.19

… ►Here $a,b,p$ are real parameters, and $k_c$ and $x$ are real or complex variables, with $p\ne 0$, $k_c\ne 0$. … ►

$k_c=k^{\prime },$

►

$p=1-{\alpha }^2,$

►

$x=\tan \varphi ,$

…

26: 11.11 Asymptotic Expansions of Anger–Weber Functions

… ►For fixed $\lambda$ $(>0)$, … ►

11.11.9_5 $$a_{k+1}(\lambda )=\frac{\lambda }{1-{\lambda }^2}\frac{\lambda a_k^{\prime \prime }(\lambda )+a_k^{\prime }(\lambda )}{(2k+1)(2k+2)},$$ $k=0,1,2,\mathrm{\dots }$.

ⓘ

Symbols:

$k$: nonnegative integer, $\lambda$: parameter and $a_k(\lambda )$: expansion function

Referenced by:

Erratum (V1.1.2) for Additions

Permalink:

http://dlmf.nist.gov/11.11.E9_5

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

… ►

11.11.12 $$\mu =\sqrt{1-{\lambda }^2}-\ln \left(\frac{1+\sqrt{1-{\lambda }^2}}{\lambda }\right),$$

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function and $\lambda$: parameter

Permalink:

http://dlmf.nist.gov/11.11.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

… ►

11.11.13_5 $${\left(\frac{1}{2}\right)}_k{b}_k(\lambda )=\frac{{(-1)}^k}{{\left(1-{\lambda }^2\right)}^{1/4}}{U}_k\left(\frac{1}{\sqrt{1-{\lambda }^2}}\right),$$ $k=0,1,2,\mathrm{\dots }$,

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $k$: nonnegative integer, $\lambda$: parameter and $b_k(\lambda )$: expansion function

Referenced by:

Erratum (V1.1.2) for Additions

Permalink:

http://dlmf.nist.gov/11.11.E13_5

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

… ►

11.11.14 $${𝐀}_{-\nu }\left(\lambda \nu \right)\sim \frac{1}{\pi \nu (\lambda -1)},$$ $\lambda >1$, $|\mathrm{ph}\nu |\le \pi -\delta$,

ⓘ

Symbols:

${𝐀}_{\nu }\left(z\right)$: Anger–Weber function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $\nu$: real or complex order, $\delta$: arbitrary small positive constant and $\lambda$: parameter

Source:

Olver (1997b, pp. 103 and 352)

Referenced by:

§11.11(iii)

Permalink:

http://dlmf.nist.gov/11.11.E14

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.2):

The constraint has been extended to include $|\mathrm{ph}\nu |\le \pi -\delta$.

Suggested 2021-04-06 by Gergő Nemes

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

…

27: 12.11 Zeros

… ►

§12.11(iii) Asymptotic Expansions for Large Parameter

… ►

12.11.4 $$u_{a,s}\sim 2^{\frac{1}{2}}\mu \left(p_0(\alpha )+\frac{p_1(\alpha )}{{\mu }^4}+\frac{p_2(\alpha )}{{\mu }^8}+\mathrm{\cdots }\right),$$

ⓘ

Defines:

$u_{a,s}$: zeros (locally)

Symbols:

$\sim$: Poincaré asymptotic expansion, $s$: nonnegative integer, $a$: real or complex parameter, $\alpha$ and $p_n(\zeta )$: coefficients

Referenced by:

§12.11(iii)

Permalink:

http://dlmf.nist.gov/12.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(iii), §12.11 and Ch.12

… ►

12.11.5 $$p_0(\zeta )=t(\zeta ),$$

ⓘ

Symbols:

$\zeta$: change of variable and $p_n(\zeta )$: coefficients

Permalink:

http://dlmf.nist.gov/12.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(iii), §12.11 and Ch.12

… ►

12.11.6 $$p_1(\zeta )=\frac{t^3-6t}{24{(t^2-1)}^2}+\frac{5}{48{((t^2-1){\zeta }^3)}^{\frac{1}{2}}}.$$

ⓘ

Symbols:

$\zeta$: change of variable and $p_n(\zeta )$: coefficients

Permalink:

http://dlmf.nist.gov/12.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(iii), §12.11 and Ch.12

… ►

12.11.8 $$q_0(\zeta )=t(\zeta ).$$

ⓘ

Symbols:

$\zeta$: change of variable

Permalink:

http://dlmf.nist.gov/12.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §12.11(iii), §12.11 and Ch.12

…

28: 28.4 Fourier Series

… ►

28.4.10 $$\sum _{m=0}^{\mathrm{\infty }}{\left(A_{2m+1}^{2n+1}(q)\right)}^2=1,$$

ⓘ

Symbols:

$m$: integer, $q=h^2$: parameter, $n$: integer and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.4.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §28.4(iii), §28.4 and Ch.28

►

28.4.11 $$\sum _{m=0}^{\mathrm{\infty }}{\left(B_{2m+1}^{2n+1}(q)\right)}^2=1,$$

ⓘ

Symbols:

$m$: integer, $q=h^2$: parameter, $n$: integer and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.4.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §28.4(iii), §28.4 and Ch.28

►

28.4.12 $$\sum _{m=0}^{\mathrm{\infty }}{\left(B_{2m+2}^{2n+2}(q)\right)}^2=1.$$

ⓘ

Symbols:

$m$: integer, $q=h^2$: parameter, $n$: integer and $B_m(q)$: Fourier coefficient

Referenced by:

item (d)

Permalink:

http://dlmf.nist.gov/28.4.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §28.4(iii), §28.4 and Ch.28

… ►

§28.4(v) Change of Sign of $q$

►

28.4.17 $$A_{2m}^{2n}(-q)={(-1)}^{n-m}{A}_{2m}^{2n}(q),$$

ⓘ

Symbols:

$m$: integer, $q=h^2$: parameter, $n$: integer and $A_m(q)$: Fourier coefficient

A&S Ref:

20.8.5

Permalink:

http://dlmf.nist.gov/28.4.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §28.4(v), §28.4 and Ch.28

…

29: 12.14 The Function $W(a,x)$

… ►These follow from the contour integrals of §12.5(ii), which are valid for general complex values of the argument $z$ and parameter $a$. … ►

§12.14(ix) Uniform Asymptotic Expansions for Large Parameter

… ►

Positive $a$,

… ►

Airy-type Uniform Expansions

… ► …

30: 25.11 Hurwitz Zeta Function

… ►

25.11.15 $$\zeta (s,ka)=k^{-s}\sum _{n=0}^{k-1}\zeta (s,a+\frac{n}{k}),$$ $s\ne 1$, $k=1,2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer, $a$: real or complex parameter and $s$: complex variable

Keywords:

special value

Proof sketch:

Derivable by starting with (25.11.1), replacing $a\mapsto ka$, pulling out a factor of $k^{-s}$ and performing a change of sum index $n=km+l$ with $m\ge 0$ and $0\le l\le k-1$, which through Euclidean division converts the single sum into a double sum of the correct form.

Referenced by:

(25.11.24)

Permalink:

http://dlmf.nist.gov/25.11.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(v), §25.11 and Ch.25

… ►

25.11.31 $$\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\frac{x^{s-1}{\mathrm{e}}^{-ax}}{2\cosh x}dx=4^{-s}\left(\zeta (s,\frac{1}{4}+\frac{1}{4}a)-\zeta (s,\frac{3}{4}+\frac{1}{4}a)\right),$$ $\mathrm{\Re }s>0$, $\mathrm{\Re }a>-1$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable, $a$: real or complex parameter and $s$: complex variable

Keywords:

improper integral, integral representation

Proof sketch:

The improper integral can be evaluated by using $2\cosh x={\mathrm{e}}^x(1+{\mathrm{e}}^{-2x})$, changing variables $2x=t$, and then applying (25.11.35).

Permalink:

http://dlmf.nist.gov/25.11.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(viii), §25.11 and Ch.25

… ►

25.11.35 $$\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n}{{(n+a)}^s}=\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\frac{x^{s-1}{\mathrm{e}}^{-ax}}{1+{\mathrm{e}}^{-x}}dx=2^{-s}\left(\zeta (s,\frac{1}{2}a)-\zeta (s,\frac{1}{2}(1+a))\right),$$ $\mathrm{\Re }a>0$, $\mathrm{\Re }s>0$; or $\mathrm{\Re }a=0$, $\mathrm{\Im }a\ne 0$, .

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part, $\int$: integral, $\mathrm{\Re }$: real part, $n$: nonnegative integer, $x$: real variable, $a$: real or complex parameter and $s$: complex variable

Keywords:

improper integral, infinite series, integral representation, series representation

Proof sketch:

The infinite series connection to the difference of Hurwitz zeta functions is a restatement of (25.11.8). The integral representation is derivable from the difference of Hurwitz zeta functions by substituting (25.11.25) in each, making a change of variables $x=2t$, factoring, and expressing $1-{\mathrm{e}}^{-2t}=(1+{\mathrm{e}}^{-t})(1-{\mathrm{e}}^{-t})$.

Referenced by:

(25.11.31), §25.11(x), §25.11(x)

Permalink:

http://dlmf.nist.gov/25.11.E35

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(x), §25.11 and Ch.25

…