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21: 4.8 Identities

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4.8.2 $$\ln \left(z_1{z}_2\right)=\ln z_1+\ln z_2,$$ $-\pi \le \mathrm{ph}{z}_1+\mathrm{ph}{z}_2\le \pi$,

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase and $z$: complex variable

A&S Ref:

4.1.7

Permalink:

http://dlmf.nist.gov/4.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.8(i), §4.8 and Ch.4

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4.8.7 $$\ln \frac{1}{z}=-\ln z,$$ $|\mathrm{ph}z|\le \pi$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase and $z$: complex variable

Referenced by:

§4.2(i), §4.8(i)

Permalink:

http://dlmf.nist.gov/4.8.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §4.8(i), §4.8 and Ch.4

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4.8.9 $$\ln \left(\exp z\right)=z,$$ $-\pi \le \mathrm{\Im }z\le \pi$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{\Im }$: imaginary part, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.2.3

Permalink:

http://dlmf.nist.gov/4.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §4.8(i), §4.8 and Ch.4

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4.8.12 $$\ln \left(a^z\right)=z\ln a+2k\pi \mathrm{i},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $k$: integer, $a$: real or complex constant and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.8.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §4.8(i), §4.8 and Ch.4

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4.8.15 $$a^z{b}^z={(ab)}^z,$$ $-\pi \le \mathrm{ph}a+\mathrm{ph}b\le \pi$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $a$: real or complex constant and $z$: complex variable

A&S Ref:

4.2.16

Permalink:

http://dlmf.nist.gov/4.8.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §4.8(ii), §4.8 and Ch.4

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22: 7.2 Definitions

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7.2.1 $$\mathrm{erf}z=\frac{2}{\sqrt{\pi }}{\int }_0^z{\mathrm{e}}^{-t^2}dt,$$

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Defines:

$\mathrm{erf}z$: error function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $z$: complex variable

A&S Ref:

7.1.1

Referenced by:

§7.5, §7.6(ii)

Permalink:

http://dlmf.nist.gov/7.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.2(i), §7.2 and Ch.7

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7.2.2 $$\mathrm{erfc}z=\frac{2}{\sqrt{\pi }}{\int }_z^{\mathrm{\infty }}{\mathrm{e}}^{-t^2}dt=1-\mathrm{erf}z,$$

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Defines:

$\mathrm{erfc}z$: complementary error function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $z$: complex variable

A&S Ref:

7.1.2

Referenced by:

§7.5, §7.7(i)

Permalink:

http://dlmf.nist.gov/7.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §7.2(i), §7.2 and Ch.7

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7.2.6 $$ℱ\left(z\right)={\int }_z^{\mathrm{\infty }}{\mathrm{e}}^{\frac{1}{2}\pi \mathrm{i}{t}^2}dt,$$

ⓘ

Defines:

$ℱ\left(z\right)$: Fresnel integral

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 $z$: complex variable

Referenced by:

§7.5, §7.7(ii)

Permalink:

http://dlmf.nist.gov/7.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §7.2(iii), §7.2 and Ch.7

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7.2.7 $$C\left(z\right)={\int }_0^z\cos \left(\frac{1}{2}\pi t^2\right)dt,$$

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Defines:

$C\left(z\right)$: Fresnel integral

Symbols:

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

A&S Ref:

7.3.1

Referenced by:

§7.5, §7.6(i)

Permalink:

http://dlmf.nist.gov/7.2.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §7.2(iii), §7.2 and Ch.7

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7.2.8 $$S\left(z\right)={\int }_0^z\sin \left(\frac{1}{2}\pi t^2\right)dt,$$

ⓘ

Defines:

$S\left(z\right)$: Fresnel integral

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function and $z$: complex variable

A&S Ref:

7.3.2

Referenced by:

§7.5, §7.6(i)

Permalink:

http://dlmf.nist.gov/7.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §7.2(iii), §7.2 and Ch.7

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23: 7.6 Series Expansions

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7.6.1 $$\mathrm{erf}z=\frac{2}{\sqrt{\pi }}\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{z}^{2n+1}}{n!(2n+1)},$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $!$: factorial (as in $n!$), $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.1.5

Referenced by:

§7.6(i)

Permalink:

http://dlmf.nist.gov/7.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.6(i), §7.6 and Ch.7

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7.6.2 $$\mathrm{erf}z=\frac{2}{\sqrt{\pi }}{\mathrm{e}}^{-z^2}\sum _{n=0}^{\mathrm{\infty }}\frac{2^n{z}^{2n+1}}{1\cdot 3\mathrm{\cdots }(2n+1)},$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.1.6

Referenced by:

§7.6(i), §7.6(ii)

Permalink:

http://dlmf.nist.gov/7.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §7.6(i), §7.6 and Ch.7

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7.6.4 $$C\left(z\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{(\frac{1}{2}\pi )}^{2n}}{(2n)!(4n+1)}{z}^{4n+1},$$

ⓘ

Symbols:

$C\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.3.11

Referenced by:

§7.6(i)

Permalink:

http://dlmf.nist.gov/7.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §7.6(i), §7.6 and Ch.7

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7.6.6 $$S\left(z\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{(\frac{1}{2}\pi )}^{2n+1}}{(2n+1)!(4n+3)}{z}^{4n+3},$$

ⓘ

Symbols:

$S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.3.13

Referenced by:

§7.6(i)

Permalink:

http://dlmf.nist.gov/7.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §7.6(i), §7.6 and Ch.7

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7.6.10 $$C\left(z\right)=z\sum _{n=0}^{\mathrm{\infty }}{𝗃}_{2n}\left(\frac{1}{2}\pi z^2\right),$$

ⓘ

Symbols:

$C\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§7.6(ii)

Permalink:

http://dlmf.nist.gov/7.6.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §7.6(ii), §7.6 and Ch.7

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24: 25.4 Reflection Formulas

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25.4.1 $$\zeta \left(1-s\right)=2{(2\pi )}^{-s}\cos \left(\frac{1}{2}\pi s\right)\mathrm{\Gamma }\left(s\right)\zeta \left(s\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function and $s$: complex variable

Keywords:

reflection formula

Source:

Apostol (1976, (12), p. 259)

Referenced by:

§25.10(i), §5.17

Permalink:

http://dlmf.nist.gov/25.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §25.4 and Ch.25

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25.4.2 $$\zeta \left(s\right)=2{(2\pi )}^{s-1}\sin \left(\frac{1}{2}\pi s\right)\mathrm{\Gamma }\left(1-s\right)\zeta \left(1-s\right).$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function and $s$: complex variable

Keywords:

reflection formula

Source:

Apostol (1976, (13), p. 259)

A&S Ref:

23.2.6

Permalink:

http://dlmf.nist.gov/25.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.4 and Ch.25

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25.4.4 $$\xi \left(s\right)=\frac{1}{2}s(s-1)\mathrm{\Gamma }\left(\frac{1}{2}s\right){\pi }^{-s/2}\zeta \left(s\right).$$

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Defines:

$\xi \left(s\right)$: Riemann’s $\xi$-function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter and $s$: complex variable

Keywords:

definition

Source:

Apostol (1976, p. 260)

Permalink:

http://dlmf.nist.gov/25.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §25.4 and Ch.25

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25.4.5 $${(-1)}^k{\zeta }^{(k)}\left(1-s\right)=\frac{2}{{(2\pi )}^s}\sum _{m=0}^k\sum _{r=0}^m\left(\genfrac{}{}{0.0pt}{}{k}{m}\right)\left(\genfrac{}{}{0.0pt}{}{m}{r}\right)\left(\mathrm{\Re }\left(c^{k-m}\right)\cos \left(\frac{1}{2}\pi s\right)+\mathrm{\Im }\left(c^{k-m}\right)\sin \left(\frac{1}{2}\pi s\right)\right){\mathrm{\Gamma }}^{(r)}\left(s\right){\zeta }^{(m-r)}\left(s\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{\Im }$: imaginary part, $\mathrm{\Re }$: real part, $\sin z$: sine function, $k$: nonnegative integer, $m$: nonnegative integer, $s$: complex variable and $c$

Keywords:

derivative

Source:

Apostol (1985a, (6), p. 224)

Permalink:

http://dlmf.nist.gov/25.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §25.4 and Ch.25

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25.4.6 $$c\equiv -\ln \left(2\pi \right)-\frac{1}{2}\pi \mathrm{i}.$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function and $c$

Keywords:

definition

Source:

Apostol (1985a, p. 223)

Referenced by:

§25.6(ii)

Permalink:

http://dlmf.nist.gov/25.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §25.4 and Ch.25

25: 7.5 Interrelations

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7.5.1 $$F\left(z\right)=\frac{1}{2}\mathrm{i}\sqrt{\pi }\left({\mathrm{e}}^{-z^2}-w\left(z\right)\right)=-\frac{1}{2}\mathrm{i}\sqrt{\pi }{\mathrm{e}}^{-z^2}\mathrm{erf}\left(\mathrm{i}z\right).$$

ⓘ

Symbols:

$F\left(z\right)$: Dawson’s integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $w\left(z\right)$: Faddeeva (or Faddeyeva) function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

Referenced by:

§7.5

Permalink:

http://dlmf.nist.gov/7.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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7.5.3 $$C\left(z\right)=\frac{1}{2}+\mathrm{f}\left(z\right)\sin \left(\frac{1}{2}\pi z^2\right)-\mathrm{g}\left(z\right)\cos \left(\frac{1}{2}\pi z^2\right),$$

ⓘ

Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $C\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function and $z$: complex variable

A&S Ref:

7.3.9

Referenced by:

§7.12(ii), §7.5

Permalink:

http://dlmf.nist.gov/7.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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7.5.5 $${\mathrm{e}}^{-\frac{1}{2}\pi \mathrm{i}{z}^2}ℱ\left(z\right)=\mathrm{g}\left(z\right)+\mathrm{i}\mathrm{f}\left(z\right).$$

ⓘ

Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $ℱ\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

Referenced by:

§7.13(iv), §7.5, §7.7(ii)

Permalink:

http://dlmf.nist.gov/7.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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7.5.7 $$\zeta =\frac{1}{2}\sqrt{\pi }(1\mp \mathrm{i})z,$$

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Defines:

$\zeta$: change of variable (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit and $z$: complex variable

Referenced by:

§7.12(ii), §7.5

Permalink:

http://dlmf.nist.gov/7.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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7.5.13 $$G\left(x\right)=\sqrt{\pi }F\left(x\right)-\frac{1}{2}{\mathrm{e}}^{-x^2}\mathrm{Ei}\left(x^2\right),$$ $x>0$.

ⓘ

Symbols:

$F\left(z\right)$: Dawson’s integral, $G\left(z\right)$: Goodwin–Staton integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{Ei}\left(x\right)$: exponential integral and $x$: real variable

Referenced by:

§7.5

Permalink:

http://dlmf.nist.gov/7.5.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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26: 11.4 Basic Properties

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11.4.5 $${𝐇}_{\frac{1}{2}}\left(z\right)={\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}(1-\cos z),$$

ⓘ

Symbols:

${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function and $z$: complex variable

A&S Ref:

12.1.16

Referenced by:

§11.4(i)

Permalink:

http://dlmf.nist.gov/11.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §11.4(i), §11.4 and Ch.11

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11.4.6 $${𝐇}_{-\frac{1}{2}}\left(z\right)={\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}\sin z,$$

ⓘ

Symbols:

${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/11.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §11.4(i), §11.4 and Ch.11

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11.4.7 $${𝐋}_{\frac{1}{2}}\left(z\right)={\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}(\cosh z-1),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, ${𝐋}_{\nu }\left(z\right)$: modified Struve function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/11.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §11.4(i), §11.4 and Ch.11

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11.4.8 $${𝐋}_{-\frac{1}{2}}\left(z\right)={\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}\sinh z,$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\sinh z$: hyperbolic sine function, ${𝐋}_{\nu }\left(z\right)$: modified Struve function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/11.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §11.4(i), §11.4 and Ch.11

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11.4.10 $${𝐇}_{-\frac{3}{2}}\left(z\right)={\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}\left(\cos z-\frac{\sin z}{z}\right),$$

ⓘ

Symbols:

${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/11.4.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §11.4(i), §11.4 and Ch.11

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27: 22.11 Fourier and Hyperbolic Series

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22.11.1 $$\mathrm{sn}(z,k)=\frac{2\pi }{Kk}\sum _{n=0}^{\mathrm{\infty }}\frac{q^{n+\frac{1}{2}}\sin \left((2n+1)\zeta \right)}{1-q^{2n+1}},$$

ⓘ

Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $q$: nome, $\sin z$: sine function, $z$: complex, $k$: modulus and $\zeta$: change of variable

A&S Ref:

16.23.1

Permalink:

http://dlmf.nist.gov/22.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

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22.11.2 $$\mathrm{cn}(z,k)=\frac{2\pi }{Kk}\sum _{n=0}^{\mathrm{\infty }}\frac{q^{n+\frac{1}{2}}\cos \left((2n+1)\zeta \right)}{1+q^{2n+1}},$$

ⓘ

Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\cos z$: cosine function, $q$: nome, $z$: complex, $k$: modulus and $\zeta$: change of variable

A&S Ref:

16.23.2

Permalink:

http://dlmf.nist.gov/22.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

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22.11.3 $$\mathrm{dn}(z,k)=\frac{\pi }{2K}+\frac{2\pi }{K}\sum _{n=1}^{\mathrm{\infty }}\frac{q^n\cos \left(2n\zeta \right)}{1+q^{2n}}.$$

ⓘ

Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\cos z$: cosine function, $q$: nome, $z$: complex, $k$: modulus and $\zeta$: change of variable

A&S Ref:

16.23.3

Permalink:

http://dlmf.nist.gov/22.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

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22.11.4 $$\mathrm{cd}(z,k)=\frac{2\pi }{Kk}\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{q}^{n+\frac{1}{2}}\cos \left((2n+1)\zeta \right)}{1-q^{2n+1}},$$

ⓘ

Symbols:

$\mathrm{cd}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\cos z$: cosine function, $q$: nome, $z$: complex, $k$: modulus and $\zeta$: change of variable

A&S Ref:

16.23.4

Permalink:

http://dlmf.nist.gov/22.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

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22.11.5 $$\mathrm{sd}(z,k)=\frac{2\pi }{Kk{k}^{\prime }}\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n{q}^{n+\frac{1}{2}}\sin \left((2n+1)\zeta \right)}{1+q^{2n+1}},$$

ⓘ

Symbols:

$\mathrm{sd}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $q$: nome, $\sin z$: sine function, $z$: complex, $k$: modulus, $k^{\prime }$: complementary modulus and $\zeta$: change of variable

A&S Ref:

16.23.5

Permalink:

http://dlmf.nist.gov/22.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

…

28: 10.34 Analytic Continuation

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10.34.1 $$I_{\nu }\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={\mathrm{e}}^{m\nu \pi \mathrm{i}}{I}_{\nu }\left(z\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $m$: integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.30

Referenced by:

§10.30(ii), §10.34, §10.47(v)

Permalink:

http://dlmf.nist.gov/10.34.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.34 and Ch.10

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10.34.2 $$K_{\nu }\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={\mathrm{e}}^{-m\nu \pi \mathrm{i}}{K}_{\nu }\left(z\right)-\pi \mathrm{i}\sin \left(m\nu \pi \right)\csc \left(\nu \pi \right)I_{\nu }\left(z\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\sin z$: sine function, $m$: integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.31

Referenced by:

§10.27, §10.34, §10.34

Permalink:

http://dlmf.nist.gov/10.34.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.34 and Ch.10

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10.34.4 $$K_{\nu }\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)=\csc \left(\nu \pi \right)\left(\pm \sin \left(m\nu \pi \right)K_{\nu }\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)\mp \sin \left((m\mp 1)\nu \pi \right)K_{\nu }\left(z\right)\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\sin z$: sine function, $m$: integer, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.34, §10.40(i)

Permalink:

http://dlmf.nist.gov/10.34.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.34 and Ch.10

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10.34.5 $$K_n\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={(-1)}^{mn}{K}_n\left(z\right)+{(-1)}^{n(m-1)-1}m\pi \mathrm{i}{I}_n\left(z\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $m$: integer, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/10.34.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.34 and Ch.10

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10.34.6 $$K_n\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)=\pm {(-1)}^{n(m-1)}m{K}_n\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)\mp {(-1)}^{nm}(m\mp 1)K_n\left(z\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $m$: integer, $n$: integer and $z$: complex variable

Referenced by:

§10.40(i)

Permalink:

http://dlmf.nist.gov/10.34.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.34 and Ch.10

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29: 11.7 Integrals and Sums

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11.7.2 $$\int z^{-\nu }{𝐇}_{\nu +1}\left(z\right)dz=-z^{-\nu }{𝐇}_{\nu }\left(z\right)+\frac{2^{-\nu }z}{\sqrt{\pi }\mathrm{\Gamma }\left(\nu +\frac{3}{2}\right)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $z$: complex variable and $\nu$: real or complex order

A&S Ref:

12.1.24

Permalink:

http://dlmf.nist.gov/11.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §11.7(i), §11.7 and Ch.11

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11.7.4 $$\int z^{-\nu }{𝐋}_{\nu +1}\left(z\right)dz=z^{-\nu }{𝐋}_{\nu }\left(z\right)-\frac{2^{-\nu }z}{\sqrt{\pi }\mathrm{\Gamma }\left(\nu +\frac{3}{2}\right)}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $z$: complex variable and $\nu$: real or complex order

Permalink:

http://dlmf.nist.gov/11.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §11.7(i), §11.7 and Ch.11

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11.7.6 $$f_{\nu +1}(z)=(2\nu +1)f_{\nu }(z)-z^{\nu +1}{𝐇}_{\nu }\left(z\right)+\frac{{(\frac{1}{2}{z}^2)}^{\nu +1}}{(\nu +1)\sqrt{\pi }\mathrm{\Gamma }\left(\nu +\frac{3}{2}\right)},$$ $\mathrm{\Re }\nu >-1$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Re }$: real part, $z$: complex variable and $\nu$: real or complex order

A&S Ref:

12.1.28 (The condition on $\nu$ has been corrected.)

Permalink:

http://dlmf.nist.gov/11.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §11.7(i), §11.7 and Ch.11

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11.7.9 $${\int }_0^{\mathrm{\infty }}{𝐇}_{\nu }\left(t\right)dt=-\cot \left(\frac{1}{2}\pi \nu \right),$$ ,

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Symbols:

${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part and $\nu$: real or complex order

Permalink:

http://dlmf.nist.gov/11.7.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §11.7(ii), §11.7 and Ch.11

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11.7.10 $${\int }_0^{\mathrm{\infty }}{t}^{-\nu -1}{𝐇}_{\nu }\left(t\right)dt=\frac{\pi }{2^{\nu +1}\mathrm{\Gamma }\left(\nu +1\right)},$$ $\mathrm{\Re }\nu >-\frac{3}{2}$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part and $\nu$: real or complex order

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/11.7.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §11.7(ii), §11.7 and Ch.11

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30: 20.3 Graphics

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