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21: 4.14 Definitions and Periodicity

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4.14.1 $$\sin z=\frac{{\mathrm{e}}^{\mathrm{i}z}-{\mathrm{e}}^{-\mathrm{i}z}}{2\mathrm{i}},$$

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Defines:

$\sin z$: sine function

Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

4.3.1

Referenced by:

§4.45(ii), (9.5.3)

Permalink:

http://dlmf.nist.gov/4.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.14 and Ch.4

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4.14.2 $$\cos z=\frac{{\mathrm{e}}^{\mathrm{i}z}+{\mathrm{e}}^{-\mathrm{i}z}}{2},$$

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Defines:

$\cos z$: cosine function

Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

4.3.2

Referenced by:

(9.5.3)

Permalink:

http://dlmf.nist.gov/4.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.14 and Ch.4

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4.14.3 $$\cos z\pm \mathrm{i}\sin z={\mathrm{e}}^{\pm \mathrm{i}z},$$

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Symbols:

$\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.14 and Ch.4

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22: 9.2 Differential Equation

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9.2.10 $$\mathrm{Bi}\left(z\right)={\mathrm{e}}^{-\pi \mathrm{i}/6}\mathrm{Ai}\left(z{\mathrm{e}}^{-2\pi \mathrm{i}/3}\right)+{\mathrm{e}}^{\pi \mathrm{i}/6}\mathrm{Ai}\left(z{\mathrm{e}}^{2\pi \mathrm{i}/3}\right).$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: 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

Proof sketch:

Derivable from Olver (1997b, p. 414).

A&S Ref:

10.4.6

Referenced by:

(9.10.19), (9.5.5)

Permalink:

http://dlmf.nist.gov/9.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §9.2(v), §9.2 and Ch.9

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9.2.12 $$\mathrm{Ai}\left(z\right)+{\mathrm{e}}^{-2\pi \mathrm{i}/3}\mathrm{Ai}\left(z{\mathrm{e}}^{-2\pi \mathrm{i}/3}\right)+{\mathrm{e}}^{2\pi \mathrm{i}/3}\mathrm{Ai}\left(z{\mathrm{e}}^{2\pi \mathrm{i}/3}\right)=0,$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: 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

Source:

Olver (1997b, (8.03), p. 414)

A&S Ref:

10.4.7

Permalink:

http://dlmf.nist.gov/9.2.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §9.2(v), §9.2 and Ch.9

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9.2.13 $$\mathrm{Bi}\left(z\right)+{\mathrm{e}}^{-2\pi \mathrm{i}/3}\mathrm{Bi}\left(z{\mathrm{e}}^{-2\pi \mathrm{i}/3}\right)+{\mathrm{e}}^{2\pi \mathrm{i}/3}\mathrm{Bi}\left(z{\mathrm{e}}^{2\pi \mathrm{i}/3}\right)=0.$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: 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

Proof sketch:

Derivable from Olver (1997b, p. 414).

A&S Ref:

10.4.8

Permalink:

http://dlmf.nist.gov/9.2.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §9.2(v), §9.2 and Ch.9

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9.2.14 $$\mathrm{Ai}\left(-z\right)={\mathrm{e}}^{\pi \mathrm{i}/3}\mathrm{Ai}\left(z{\mathrm{e}}^{\pi \mathrm{i}/3}\right)+{\mathrm{e}}^{-\pi \mathrm{i}/3}\mathrm{Ai}\left(z{\mathrm{e}}^{-\pi \mathrm{i}/3}\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: 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

Source:

Olver (1997b, p. 414)

Permalink:

http://dlmf.nist.gov/9.2.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §9.2(v), §9.2 and Ch.9

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9.2.15 $$\mathrm{Bi}\left(-z\right)={\mathrm{e}}^{-\pi \mathrm{i}/6}\mathrm{Ai}\left(z{\mathrm{e}}^{\pi \mathrm{i}/3}\right)+{\mathrm{e}}^{\pi \mathrm{i}/6}\mathrm{Ai}\left(z{\mathrm{e}}^{-\pi \mathrm{i}/3}\right).$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: 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

Source:

Olver (1997b, p. 414)

Permalink:

http://dlmf.nist.gov/9.2.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §9.2(v), §9.2 and Ch.9

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23: 36 Integrals with Coalescing Saddles

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24: 5.13 Integrals

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5.13.1 $$\frac{1}{2\pi \mathrm{i}}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}\mathrm{\Gamma }\left(s+a\right)\mathrm{\Gamma }\left(b-s\right)z^{-s}ds=\frac{\mathrm{\Gamma }\left(a+b\right)z^a}{{(1+z)}^{a+b}},$$ $\mathrm{\Re }\left(a+b\right)>0$, , .

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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 $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $s$: real or complex variable, $z$: complex variable, $a$: real or complex variable and $b$: real or complex variable

Referenced by:

§5.13, §5.13, §5.19(ii)

Permalink:

http://dlmf.nist.gov/5.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §5.13 and Ch.5

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5.13.2 $$\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|\mathrm{\Gamma }\left(a+\mathrm{i}t\right)|}^2{\mathrm{e}}^{(2b-\pi )t}dt=\frac{\mathrm{\Gamma }\left(2a\right)}{{(2\sin b)}^{2a}},$$ $a>0$, .

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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 $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $a$: real or complex variable and $b$: real or complex variable

Referenced by:

§5.13

Permalink:

http://dlmf.nist.gov/5.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.13 and Ch.5

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5.13.3 $$\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{\Gamma }\left(a+\mathrm{i}t\right)\mathrm{\Gamma }\left(b+\mathrm{i}t\right)\mathrm{\Gamma }\left(c-\mathrm{i}t\right)\mathrm{\Gamma }\left(d-\mathrm{i}t\right)dt=\frac{\mathrm{\Gamma }\left(a+c\right)\mathrm{\Gamma }\left(a+d\right)\mathrm{\Gamma }\left(b+c\right)\mathrm{\Gamma }\left(b+d\right)}{\mathrm{\Gamma }\left(a+b+c+d\right)},$$ $\mathrm{\Re }a,\mathrm{\Re }b,\mathrm{\Re }c,\mathrm{\Re }d>0$.

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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 $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $a$: real or complex variable and $b$: real or complex variable

Referenced by:

§5.13

Permalink:

http://dlmf.nist.gov/5.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.13, §5.13 and Ch.5

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5.13.5 $\mathrm{\Re }\left(a_k\right)>0$, $k=1,2,3,4$.

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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 $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $j$: nonnegative integer, $k$: nonnegative integer and $a$: real or complex variable

Referenced by:

§5.13

Permalink:

http://dlmf.nist.gov/5.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.13, §5.13 and Ch.5

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25: 5.3 Graphics

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Figure 5.3.4: $|\mathrm{\Gamma }\left(x+\mathrm{i}y\right)|$. Magnify 3D Help

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Figure 5.3.5: $1/|\mathrm{\Gamma }\left(x+\mathrm{i}y\right)|$. Magnify 3D Help

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Figure 5.3.6: $|\psi \left(x+\mathrm{i}y\right)|$. Magnify 3D Help

26: 10.56 Generating Functions

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10.56.3 $$\frac{\cosh \sqrt{z^2+2\mathrm{i}zt}}{z}=\frac{\cosh z}{z}+\sum _{n=1}^{\mathrm{\infty }}\frac{{(\mathrm{i}t)}^n}{n!}{𝗂}_{n-1}^{(1)}\left(z\right),$$

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Symbols:

$!$: factorial (as in $n!$), $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable

Referenced by:

§10.56

Permalink:

http://dlmf.nist.gov/10.56.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.56 and Ch.10

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10.56.4 $$\frac{\sinh \sqrt{z^2+2\mathrm{i}zt}}{z}=\frac{\sinh z}{z}+\sum _{n=1}^{\mathrm{\infty }}\frac{{(\mathrm{i}t)}^n}{n!}{𝗂}_{n-1}^{(2)}\left(z\right),$$

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Symbols:

$!$: factorial (as in $n!$), $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, ${𝗂}_n^{(2)}\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable

Referenced by:

§10.56

Permalink:

http://dlmf.nist.gov/10.56.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.56 and Ch.10

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10.56.5 $$\frac{\exp \left(-\sqrt{z^2+2\mathrm{i}zt}\right)}{z}=\frac{{\mathrm{e}}^{-z}}{z}+\frac{2}{\pi }\sum _{n=1}^{\mathrm{\infty }}\frac{{(-\mathrm{i}t)}^n}{n!}{𝗄}_{n-1}\left(z\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, ${𝗄}_n\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable

Referenced by:

§10.56, Ch.10

Permalink:

http://dlmf.nist.gov/10.56.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.56 and Ch.10

27: 11.8 Analogs to Kelvin Functions

§11.8 Analogs to Kelvin Functions

►For properties of Struve functions of argument $x{\mathrm{e}}^{\pm 3\pi \mathrm{i}/4}$ see McLachlan and Meyers (1936).

28: 25.13 Periodic Zeta Function

… ►The notation $F(x,s)$ is used for the polylogarithm ${\mathrm{Li}}_s\left({\mathrm{e}}^{2\pi \mathrm{i}x}\right)$ with $x$ real: ►

25.13.1 $$F(x,s)\equiv \sum _{n=1}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{2\pi \mathrm{i}nx}}{n^s},$$

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Defines:

$F(x,s)$: periodic zeta function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

definition, infinite series, series representation

Source:

Apostol (1976, (9), p. 257)

Permalink:

http://dlmf.nist.gov/25.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §25.13 and Ch.25

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25.13.2 $$F(x,s)=\frac{\mathrm{\Gamma }\left(1-s\right)}{{(2\pi )}^{1-s}}\left({\mathrm{e}}^{\pi \mathrm{i}(1-s)/2}\zeta (1-s,x)+{\mathrm{e}}^{\pi \mathrm{i}(s-1)/2}\zeta (1-s,1-x)\right),$$ , $\mathrm{\Re }s>1$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $F(x,s)$: periodic zeta function, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable

Keywords:

connection formula

Source:

Apostol (1976, Exercise 2, p. 273)

Permalink:

http://dlmf.nist.gov/25.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.13 and Ch.25

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25.13.3 $$\zeta (1-s,x)=\frac{\mathrm{\Gamma }\left(s\right)}{{(2\pi )}^s}\left({\mathrm{e}}^{-\pi \mathrm{i}s/2}F(x,s)+{\mathrm{e}}^{\pi \mathrm{i}s/2}F(-x,s)\right),$$ $\mathrm{\Re }s>0$ if ; $\mathrm{\Re }s>1$ if $x=1$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $F(x,s)$: periodic zeta function, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable

Keywords:

connection formula

Source:

Apostol (1976, (10), p. 257)

Notes:

This formula is equivalent to (25.11.9).

Referenced by:

Erratum (V1.1.4) for Equation (25.13.3)

Permalink:

http://dlmf.nist.gov/25.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.13 and Ch.25

29: 23.17 Elementary Properties

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$\lambda \left(\mathrm{i}\right)=\frac{1}{2},$

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$\lambda \left({\mathrm{e}}^{\pi \mathrm{i}/3}\right)={\mathrm{e}}^{\pi \mathrm{i}/3},$

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$J\left(\mathrm{i}\right)=1,$

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$J\left({\mathrm{e}}^{\pi \mathrm{i}/3}\right)=0,$

… ►with $q^{1/12}={\mathrm{e}}^{\mathrm{i}\pi \tau /12}$.

30: 22.4 Periods, Poles, and Zeros

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§22.4(ii) Graphical Interpretation via Glaisher’s Notation

►Figure 22.4.2 depicts the fundamental unit cell in the $z$-plane, with vertices $\text{s}=0$, $\text{c}=K$, $\text{d}=K+\mathrm{i}{K}^{\prime }$, $\text{n}=\mathrm{i}{K}^{\prime }$. The set of points $z=mK+n\mathrm{i}{K}^{\prime }$, $m,n\in \mathbb{Z}$, comprise the lattice for the 12 Jacobian functions; all other lattice unit cells are generated by translation of the fundamental unit cell by $mK+n\mathrm{i}{K}^{\prime }$, where again $m,n\in \mathbb{Z}$. … ►Let p,q be any two distinct letters from the set s,c,d,n which appear in counterclockwise orientation at the corners of all lattice unit cells. Then: (a) In any lattice unit cell $\mathrm{p}\mathrm{q}(z,k)$ has a simple zero at $z=\text{p}$ and a simple pole at $z=\text{q}$. …