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51: 9.11 Products

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9.11.2

\mathscr{W}\left\{{\operatorname{Ai}}^{2}\left(z\right),\operatorname{Ai}\left% (z\right)\operatorname{Bi}\left(z\right),{\operatorname{Bi}}^{2}\left(z\right)% \right\}=2\pi^{-3}.

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\mathscr{W}$ : Wronskian, $\pi$ : the ratio of the circumference of a circle to its diameter and $z$ : complex variable
Proof sketch:: Use (9.2.1) and (9.2.7).
Permalink:: http://dlmf.nist.gov/9.11.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §9.11(ii), §9.11 and Ch.9

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9.11.12

\int\frac{\,\mathrm{d}z}{{\operatorname{Ai}}^{2}\left(z\right)}=\pi\frac{% \operatorname{Bi}\left(z\right)}{\operatorname{Ai}\left(z\right)},

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $z$ : complex variable
Source:: Albright and Gavathas (1986, (8))
Permalink:: http://dlmf.nist.gov/9.11.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §9.11(iv), §9.11(iv), §9.11 and Ch.9

►

9.11.13

\int\frac{\,\mathrm{d}z}{\operatorname{Ai}\left(z\right)\operatorname{Bi}\left% (z\right)}=\pi\ln\left(\frac{\operatorname{Bi}\left(z\right)}{\operatorname{Ai% }\left(z\right)}\right),

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable
Source:: Albright and Gavathas (1986, (10))
Permalink:: http://dlmf.nist.gov/9.11.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §9.11(iv), §9.11(iv), §9.11 and Ch.9

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9.11.16

\int_{-\infty}^{\infty}{\operatorname{Ai}}^{3}\left(t\right)\,\mathrm{d}t=% \frac{{\Gamma}^{2}\left(\frac{1}{3}\right)}{4\pi^{2}},

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ and $\int$ : integral
Source:: Reid (1997a, (2.17))
Permalink:: http://dlmf.nist.gov/9.11.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §9.11(v), §9.11 and Ch.9

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9.11.18

\int_{0}^{\infty}{\operatorname{Ai}}^{4}\left(t\right)\,\mathrm{d}t=\frac{\ln 3% }{24\pi^{2}}.

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\ln\NVar{z}$ : principal branch of logarithm function
Source:: Laurenzi (1993, p. 905)
Permalink:: http://dlmf.nist.gov/9.11.E18
Encodings:: pMML, png, TeX
See also:: Annotations for §9.11(v), §9.11 and Ch.9

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52: 21.2 Definitions

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21.2.1

\theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)=\sum_{\mathbf{n}\in% {\mathbb{Z}}^{g}}e^{2\pi i\left(\frac{1}{2}\mathbf{n}\cdot\boldsymbol{{\Omega}% }\cdot\mathbf{n}+\mathbf{n}\cdot\mathbf{z}\right)}.

ⓘ

Defines:: $\theta\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\mathbb{Z}$ : set of all integers, $g$ : positive integer and $\boldsymbol{{\Omega}}$ : a Riemann matrix
Referenced by:: §21.2(ii)
Permalink:: http://dlmf.nist.gov/21.2.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §21.2(i), §21.2 and Ch.21

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21.2.8

\theta\left(z\middle|\Omega\right)=\theta_{3}\left(\pi z\middle|\Omega\right),

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $\theta\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function and $\pi$ : the ratio of the circumference of a circle to its diameter
Permalink:: http://dlmf.nist.gov/21.2.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §21.2(iii), §21.2 and Ch.21

►

21.2.9

\theta_{1}\left(\pi z\middle|\Omega\right)=-\theta\genfrac{[}{]}{0.0pt}{}{% \frac{1}{2}}{\frac{1}{2}}\left(z\middle|\Omega\right),

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics and $\pi$ : the ratio of the circumference of a circle to its diameter
Permalink:: http://dlmf.nist.gov/21.2.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §21.2(iii), §21.2 and Ch.21

►

21.2.10

\theta_{2}\left(\pi z\middle|\Omega\right)=\theta\genfrac{[}{]}{0.0pt}{}{% \tfrac{1}{2}}{0}\left(z\middle|\Omega\right),

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics and $\pi$ : the ratio of the circumference of a circle to its diameter
Permalink:: http://dlmf.nist.gov/21.2.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §21.2(iii), §21.2 and Ch.21

►

21.2.11

\theta_{3}\left(\pi z\middle|\Omega\right)=\theta\genfrac{[}{]}{0.0pt}{}{0}{0}% \left(z\middle|\Omega\right),

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics and $\pi$ : the ratio of the circumference of a circle to its diameter
Permalink:: http://dlmf.nist.gov/21.2.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §21.2(iii), §21.2 and Ch.21

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53: 7.14 Integrals

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7.14.2

\int_{0}^{\infty}e^{-at}\operatorname{erf}\left(bt\right)\,\mathrm{d}t=\frac{1% }{a}e^{a^{2}/(4b^{2})}\operatorname{erfc}\left(\frac{a}{2b}\right),

\Re a>0

,

|\operatorname{ph}b|<\tfrac{1}{4}\pi

,

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$ : complementary error function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{erf}\NVar{z}$ : error function, $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{ph}$ : phase and $\Re$ : real part
Keywords:: Laplace transform
A&S Ref:: 7.4.17
Referenced by:: §7.14(i)
Permalink:: http://dlmf.nist.gov/7.14.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §7.14(i), §7.14(i), §7.14 and Ch.7

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7.14.4

\int_{0}^{\infty}e^{(a-b)t}\operatorname{erfc}\left(\sqrt{at}+\sqrt{\frac{c}{t% }}\right)\,\mathrm{d}t=\frac{e^{-2(\sqrt{ac}+\sqrt{bc})}}{\sqrt{b}(\sqrt{a}+% \sqrt{b})},

|\operatorname{ph}a|<\frac{1}{2}\pi

,

\Re b>0

,

\Re c\geq 0

.

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$ : complementary error function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{ph}$ : phase and $\Re$ : real part
Keywords:: Laplace transform
A&S Ref:: 7.4.21
Referenced by:: §3.5(ix), §7.14(i)
Permalink:: http://dlmf.nist.gov/7.14.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §7.14(i), §7.14(i), §7.14 and Ch.7

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7.14.5

\int_{0}^{\infty}e^{-at}C\left(t\right)\,\mathrm{d}t=\frac{1}{a}\mathrm{f}% \left(\frac{a}{\pi}\right),

\Re a>0

,

ⓘ

Symbols:: $\mathrm{f}\left(\NVar{z}\right)$ : auxiliary function for Fresnel integrals, $C\left(\NVar{z}\right)$ : Fresnel integral, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and $\Re$ : real part
Keywords:: Laplace transform
A&S Ref:: 7.4.27 (in different notation)
Referenced by:: §7.14(ii)
Permalink:: http://dlmf.nist.gov/7.14.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §7.14(ii), §7.14(ii), §7.14 and Ch.7

►

7.14.6

\int_{0}^{\infty}e^{-at}S\left(t\right)\,\mathrm{d}t=\frac{1}{a}\mathrm{g}% \left(\frac{a}{\pi}\right),

\Re a>0

,

ⓘ

Symbols:: $\mathrm{g}\left(\NVar{z}\right)$ : auxiliary function for Fresnel integrals, $S\left(\NVar{z}\right)$ : Fresnel integral, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and $\Re$ : real part
Keywords:: Laplace transform
A&S Ref:: 7.4.28 (in different notation)
Referenced by:: §7.14(ii)
Permalink:: http://dlmf.nist.gov/7.14.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §7.14(ii), §7.14(ii), §7.14 and Ch.7

►

7.14.7

\int_{0}^{\infty}e^{-at}C\left(\sqrt{\frac{2t}{\pi}}\right)\,\mathrm{d}t=\frac% {(\sqrt{a^{2}+1}+a)^{\frac{1}{2}}}{2a\sqrt{a^{2}+1}},

\Re a>0

,

ⓘ

Symbols:: $C\left(\NVar{z}\right)$ : Fresnel integral, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and $\Re$ : real part
Keywords:: Laplace transform
A&S Ref:: 7.4.29 in modified form
Referenced by:: §7.14(ii)
Permalink:: http://dlmf.nist.gov/7.14.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §7.14(ii), §7.14(ii), §7.14 and Ch.7

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54: 36.7 Zeros

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36.7.3

\frac{3\pi(8n+5)}{9+8\xi_{n}}\xi_{n}^{3/2}=\dfrac{27}{16}\left(\dfrac{3}{2}% \right)^{1/2}\left(\ln\left(\frac{1}{\xi_{n}}\right)+3\ln\left(\dfrac{3}{2}% \right)\right).

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$ : principal branch of logarithm function and $m$ : integer
Referenced by:: §36.7(ii)
Permalink:: http://dlmf.nist.gov/36.7.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §36.7(ii), §36.7 and Ch.36

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36.7.4

z_{n}=\pm 3(\tfrac{1}{4}\pi(2n-\tfrac{1}{2}))^{1/3}\\ =3.48734(n-\tfrac{1}{4})^{1/3},

n=1,2,3,\dots

.

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $z$ : real parameter and $m$ : integer
Permalink:: http://dlmf.nist.gov/36.7.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §36.7(iii), §36.7 and Ch.36

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36.7.6

\exp\left(-2\pi i\left(\frac{z-z_{n}}{\Delta z}+\frac{2x}{\Delta x}\right)% \right)\*{\left(2\exp\left(\frac{-6\pi ix}{\Delta x}\right)\cos\left(\frac{2% \sqrt{3}\pi y}{\Delta x}\right)+1\right)}=\sqrt{3}.

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $y$ : real parameter, $z$ : real parameter, $m$ : integer and $x$ : real parameter
Permalink:: http://dlmf.nist.gov/36.7.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §36.7(iii), §36.7 and Ch.36

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36.7.8

r=3\left(\frac{(2n-1)\pi}{4|\sin\left(\tfrac{3}{2}\theta\right)|}\right)^{2/3}% (1+O\left(n^{-1}\right)),

n\to\infty

.

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$ : sine function and $m$ : integer
Permalink:: http://dlmf.nist.gov/36.7.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §36.7(iii), §36.7 and Ch.36

…

55: 36.11 Leading-Order Asymptotics

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36.11.3

\Psi_{2}\left(0,y\right)=\begin{cases}\sqrt{\ifrac{\pi}{y}}\left(\exp\left(% \tfrac{1}{4}i\pi\right)+o\left(1\right)\right),&y\to+\infty,\\ \sqrt{\ifrac{\pi}{|y|}}\exp\left(-\tfrac{1}{4}i\pi\right)\left(1+i\sqrt{2}\exp% \left(-\frac{1}{4}iy^{2}\right)+o\left(1\right)\right),&y\to-\infty.\end{cases}

ⓘ

Symbols:: $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $o\left(\NVar{x}\right)$ : order less than and $y$ : real parameter
Permalink:: http://dlmf.nist.gov/36.11.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §36.11, §36.11 and Ch.36

►

36.11.4

\Psi_{3}\left(x,0,0\right)=\frac{\sqrt{2\pi}}{(5|x|^{3})^{1/8}}\begin{cases}% \exp\left(-2\sqrt{2}(\ifrac{x}{5})^{5/4}\right)\left(\cos\left(2\sqrt{2}(% \ifrac{x}{5})^{5/4}-\tfrac{1}{8}\pi\right)+o\left(1\right)\right),&x\to+\infty% ,\\ \cos\left(4(\ifrac{|x|}{5})^{5/4}-\tfrac{1}{4}\pi\right)+o\left(1\right),&x\to% -\infty.\end{cases}

ⓘ

Symbols:: $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\exp\NVar{z}$ : exponential function, $o\left(\NVar{x}\right)$ : order less than and $x$ : real parameter
Referenced by:: §36.11
Permalink:: http://dlmf.nist.gov/36.11.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §36.11, §36.11 and Ch.36

►

36.11.5

\Psi_{3}\left(0,y,0\right)=\overline{\Psi_{3}(0,-y,0)}=\exp\left(\tfrac{1}{4}i% \pi\right)\sqrt{\ifrac{\pi}{y}}\left(1-(i/{\sqrt{3}})\exp\left(\tfrac{3}{2}i(% \ifrac{2y}{5})^{5/3}\right)+o\left(1\right)\right),

y\to+\infty

.

ⓘ

Symbols:: $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$ : complex conjugate, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $o\left(\NVar{x}\right)$ : order less than and $y$ : real parameter
Permalink:: http://dlmf.nist.gov/36.11.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §36.11, §36.11 and Ch.36

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36.11.7

\Psi^{(\mathrm{E})}\left(0,0,z\right)=\frac{\pi}{z}\left(i+\sqrt{3}\exp\left(% \frac{4}{27}iz^{3}\right)+o\left(1\right)\right),

z\to\pm\infty

,

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\Psi^{(\mathrm{E})}\left(\NVar{\mathbf{x}}\right)$ : elliptic umbilic canonical integral function, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $o\left(\NVar{x}\right)$ : order less than and $z$ : real parameter
Permalink:: http://dlmf.nist.gov/36.11.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §36.11, §36.11 and Ch.36

►

36.11.8

\Psi^{(\mathrm{H})}\left(0,0,z\right)=\frac{2\pi}{z}\left(1-\frac{i}{\sqrt{3}}% \exp\left(\frac{1}{27}iz^{3}\right)+o\left(1\right)\right),

z\to\pm\infty

.

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $\Psi^{(\mathrm{H})}\left(\NVar{\mathbf{x}}\right)$ : hyperbolic umbilic canonical integral function, $\mathrm{i}$ : imaginary unit, $o\left(\NVar{x}\right)$ : order less than and $z$ : real parameter
Permalink:: http://dlmf.nist.gov/36.11.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §36.11, §36.11 and Ch.36

56: 28.28 Integrals, Integral Representations, and Integral Equations

… ►

28.28.2

\dfrac{1}{2\pi}\int_{0}^{2\pi}e^{2\mathrm{i}hw}\operatorname{ce}_{n}\left(t,h^% {2}\right)\,\mathrm{d}t={\mathrm{i}}^{n}\operatorname{ce}_{n}\left(\alpha,h^{2% }\right){\operatorname{Mc}^{(1)}_{n}}\left(z,h\right),

ⓘ

Symbols:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, ${\operatorname{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, $h$ : parameter, $n$ : integer, $z$ : complex variable, $w(z)$ : Mathieu’s equation solution and $\alpha$ : parameter
A&S Ref:: 20.7.34 20.7.35 20.7.36 20.7.37
Permalink:: http://dlmf.nist.gov/28.28.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §28.28(i), §28.28 and Ch.28

►

28.28.3

\dfrac{1}{2\pi}\int_{0}^{2\pi}e^{2\mathrm{i}hw}\operatorname{se}_{n}\left(t,h^% {2}\right)\,\mathrm{d}t={\mathrm{i}}^{n}\operatorname{se}_{n}\left(\alpha,h^{2% }\right){\operatorname{Ms}^{(1)}_{n}}\left(z,h\right),

ⓘ

Symbols:: $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, ${\operatorname{Ms}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, $h$ : parameter, $n$ : integer, $z$ : complex variable, $w(z)$ : Mathieu’s equation solution and $\alpha$ : parameter
Permalink:: http://dlmf.nist.gov/28.28.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §28.28(i), §28.28 and Ch.28

►

28.28.4

\dfrac{\mathrm{i}h}{\pi}\int_{0}^{2\pi}\frac{\partial w}{\partial\alpha}e^{2% \mathrm{i}hw}\operatorname{ce}_{n}\left(t,h^{2}\right)\,\mathrm{d}t={\mathrm{i% }}^{n}\operatorname{ce}_{n}'\left(\alpha,h^{2}\right){\operatorname{Mc}^{(1)}_% {n}}\left(z,h\right),

►

28.28.5

\dfrac{\mathrm{i}h}{\pi}\int_{0}^{2\pi}\frac{\partial w}{\partial\alpha}e^{2% \mathrm{i}hw}\operatorname{se}_{n}\left(t,h^{2}\right)\,\mathrm{d}t={\mathrm{i% }}^{n}\operatorname{se}_{n}'\left(\alpha,h^{2}\right){\operatorname{Ms}^{(1)}_% {n}}\left(z,h\right).

… ►

28.28.10

0<\operatorname{ph}\left(h(\cosh z\pm 1)\right)<\pi.

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cosh\NVar{z}$ : hyperbolic cosine function, $\operatorname{ph}$ : phase, $h$ : parameter and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/28.28.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §28.28(i), §28.28 and Ch.28

…

57: 20.2 Definitions and Periodic Properties

… ►

20.2.5

z_{m,n}=(m+n\tau)\pi,

m,n\in\mathbb{Z}

,

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\in$ : element of, $\mathbb{Z}$ : set of all integers, $m$ : integer, $n$ : integer, $z$ : complex and $\tau$ : lattice parameter
A&S Ref:: 16.33.1 (in different notation)
Referenced by:: §20.6
Permalink:: http://dlmf.nist.gov/20.2.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §20.2(ii), §20.2 and Ch.20

… ►

20.2.6

\theta_{1}\left(z+(m+n\tau)\pi\middle|\tau\right)=(-1)^{m+n}q^{-n^{2}}e^{-2inz% }\theta_{1}\left(z\middle|\tau\right),

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $m$ : integer, $n$ : integer, $z$ : complex, $\tau$ : lattice parameter and $q$ : nome
A&S Ref:: 16.33.2 (in different notation)
Referenced by:: §23.6(i)
Permalink:: http://dlmf.nist.gov/20.2.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §20.2(ii), §20.2 and Ch.20

►

20.2.7

\theta_{2}\left(z+(m+n\tau)\pi\middle|\tau\right)=(-1)^{m}q^{-n^{2}}e^{-2inz}% \theta_{2}\left(z\middle|\tau\right),

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $m$ : integer, $n$ : integer, $z$ : complex, $\tau$ : lattice parameter and $q$ : nome
A&S Ref:: 16.33.3 (in different notation)
Permalink:: http://dlmf.nist.gov/20.2.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §20.2(ii), §20.2 and Ch.20

►

20.2.8

\theta_{3}\left(z+(m+n\tau)\pi\middle|\tau\right)=q^{-n^{2}}e^{-2inz}\theta_{3% }\left(z\middle|\tau\right),

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $m$ : integer, $n$ : integer, $z$ : complex, $\tau$ : lattice parameter and $q$ : nome
A&S Ref:: 16.33.4 (in different notation)
Permalink:: http://dlmf.nist.gov/20.2.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §20.2(ii), §20.2 and Ch.20

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20.2.10

M\equiv M(z|\tau)=e^{iz+(i\pi\tau/4)},

ⓘ

Defines:: $M$ (locally)
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, $z$ : complex and $\tau$ : lattice parameter
A&S Ref:: 16.33.1 (in different notation)
Permalink:: http://dlmf.nist.gov/20.2.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §20.2(iii), §20.2 and Ch.20

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58: 28.10 Integral Equations

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28.10.1

\frac{2}{\pi}\int_{0}^{\ifrac{\pi}{2}}\cos\left(2h\cos z\cos t\right)% \operatorname{ce}_{2n}\left(t,h^{2}\right)\,\mathrm{d}t=\frac{A_{0}^{2n}(h^{2}% )}{\operatorname{ce}_{2n}\left(\frac{1}{2}\pi,h^{2}\right)}\operatorname{ce}_{% 2n}\left(z,h^{2}\right),

ⓘ

Symbols:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $h$ : parameter, $n$ : integer, $z$ : complex variable and $A_{m}(q)$ : Fourier coefficient
A&S Ref:: 20.7.20 (in different form)
Permalink:: http://dlmf.nist.gov/28.10.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §28.10(i), §28.10 and Ch.28

►

28.10.2

\frac{2}{\pi}\int_{0}^{\ifrac{\pi}{2}}\cosh\left(2h\sin z\sin t\right)% \operatorname{ce}_{2n}\left(t,h^{2}\right)\,\mathrm{d}t=\frac{A_{0}^{2n}(h^{2}% )}{\operatorname{ce}_{2n}\left(0,h^{2}\right)}\operatorname{ce}_{2n}\left(z,h^% {2}\right),

ⓘ

Symbols:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\int$ : integral, $\sin\NVar{z}$ : sine function, $h$ : parameter, $n$ : integer, $z$ : complex variable and $A_{m}(q)$ : Fourier coefficient
A&S Ref:: 20.7.21 (in different form)
Permalink:: http://dlmf.nist.gov/28.10.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §28.10(i), §28.10 and Ch.28

►

28.10.3

\frac{2}{\pi}\int_{0}^{\ifrac{\pi}{2}}\sin\left(2h\cos z\cos t\right)% \operatorname{ce}_{2n+1}\left(t,h^{2}\right)\,\mathrm{d}t=-\frac{hA_{1}^{2n+1}% (h^{2})}{\operatorname{ce}_{2n+1}'\left(\frac{1}{2}\pi,h^{2}\right)}% \operatorname{ce}_{2n+1}\left(z,h^{2}\right),

ⓘ

Symbols:: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $h$ : parameter, $n$ : integer, $z$ : complex variable and $A_{m}(q)$ : Fourier coefficient
A&S Ref:: 20.7.20 (in different form)
Permalink:: http://dlmf.nist.gov/28.10.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §28.10(i), §28.10 and Ch.28

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28.10.5

\frac{2}{\pi}\int_{0}^{\ifrac{\pi}{2}}\sinh\left(2h\sin z\sin t\right)% \operatorname{se}_{2n+1}\left(t,h^{2}\right)\,\mathrm{d}t=\frac{hB_{1}^{2n+1}(% h^{2})}{\operatorname{se}_{2n+1}'\left(0,h^{2}\right)}\operatorname{se}_{2n+1}% \left(z,h^{2}\right),

ⓘ

Symbols:: $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $\sin\NVar{z}$ : sine function, $h$ : parameter, $n$ : integer, $z$ : complex variable and $B_{m}(q)$ : Fourier coefficient
A&S Ref:: 20.7.23 (in different form)
Permalink:: http://dlmf.nist.gov/28.10.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §28.10(i), §28.10 and Ch.28

… ►

28.10.10

\int_{0}^{\pi}J_{0}\left(2\sqrt{q}(\cos\tau+\cos\zeta)\right)\operatorname{ce}% _{n}\left(\tau,q\right)\,\mathrm{d}\tau=w_{\mbox{\tiny II}}(\pi;a_{n}\left(q% \right),q)\operatorname{ce}_{n}\left(\zeta,q\right).

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $q=h^{2}$ : parameter, $n$ : integer and $w_{\mbox{\tiny II}}(z;a,q)$ : fundamental solution
Permalink:: http://dlmf.nist.gov/28.10.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §28.10(ii), §28.10 and Ch.28

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59: 12.2 Differential Equations

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12.2.6

U\left(a,0\right)=\frac{\sqrt{\pi}}{2^{\frac{1}{2}a+\frac{1}{4}}\Gamma\left(% \frac{3}{4}+\frac{1}{2}a\right)},

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function and $a$ : real or complex parameter
A&S Ref:: 19.3.5
Referenced by:: §12.4
Permalink:: http://dlmf.nist.gov/12.2.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §12.2(ii), §12.2 and Ch.12

►

12.2.7

U'\left(a,0\right)=-\frac{\sqrt{\pi}}{2^{\frac{1}{2}a-\frac{1}{4}}\Gamma\left(% \frac{1}{4}+\frac{1}{2}a\right)},

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function and $a$ : real or complex parameter
A&S Ref:: 19.3.5
Permalink:: http://dlmf.nist.gov/12.2.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §12.2(ii), §12.2 and Ch.12

►

12.2.8

V\left(a,0\right)=\frac{\pi 2^{\frac{1}{2}a+\frac{1}{4}}}{\left(\Gamma\left(% \frac{3}{4}-\frac{1}{2}a\right)\right)^{2}\Gamma\left(\frac{1}{4}+\frac{1}{2}a% \right)},

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $V\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function and $a$ : real or complex parameter
A&S Ref:: 19.3.6
Permalink:: http://dlmf.nist.gov/12.2.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §12.2(ii), §12.2 and Ch.12

… ►

12.2.10

\mathscr{W}\left\{U\left(a,z\right),V\left(a,z\right)\right\}=\sqrt{2/\pi},

ⓘ

Symbols:: $\mathscr{W}$ : Wronskian, $\pi$ : the ratio of the circumference of a circle to its diameter, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $V\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter
A&S Ref:: 19.4.1
Permalink:: http://dlmf.nist.gov/12.2.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §12.2(iii), §12.2 and Ch.12

►

12.2.11

\mathscr{W}\left\{U\left(a,z\right),U\left(a,-z\right)\right\}=\frac{\sqrt{2% \pi}}{\Gamma\left(\frac{1}{2}+a\right)},

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathscr{W}$ : Wronskian, $\pi$ : the ratio of the circumference of a circle to its diameter, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/12.2.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §12.2(iii), §12.2 and Ch.12

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60: 9.12 Scorer Functions

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9.12.1

\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-zw=\frac{1}{\pi}.

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable and $w$ : function
Source:: Olver (1997b, (12.07), p. 430)
Referenced by:: (9.12.20), §9.12(i), §9.17(ii)
Permalink:: http://dlmf.nist.gov/9.12.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §9.12(i), §9.12 and Ch.9

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9.12.8

-\operatorname{Gi}\left(z\right),\operatorname{Ai}\left(z\right),\operatorname% {Bi}\left(z\right),

|\operatorname{ph}z|\leq\tfrac{1}{3}\pi

,

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\operatorname{Gi}\left(\NVar{z}\right)$ : Scorer function (inhomogeneous Airy function), $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase and $z$ : complex variable
Source:: Olver (1997b, p. 432)
Permalink:: http://dlmf.nist.gov/9.12.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §9.12(iv), §9.12 and Ch.9

►

9.12.9

\operatorname{Hi}\left(z\right),\operatorname{Ai}\left(ze^{-2\pi i/3}\right),% \operatorname{Ai}\left(ze^{2\pi i/3}\right),

|\operatorname{ph}\left(-z\right)|\leq\tfrac{2}{3}\pi

,

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\operatorname{Hi}\left(\NVar{z}\right)$ : Scorer function (inhomogeneous Airy function), $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\operatorname{ph}$ : phase and $z$ : complex variable
Source:: Olver (1997b, p. 432)
Permalink:: http://dlmf.nist.gov/9.12.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §9.12(iv), §9.12 and Ch.9

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9.12.19

\operatorname{Gi}\left(x\right)=\frac{1}{\pi}\int_{0}^{\infty}\sin\left(\tfrac% {1}{3}t^{3}+xt\right)\,\mathrm{d}t,

x\in\mathbb{R}

.

ⓘ

Symbols:: $\operatorname{Gi}\left(\NVar{z}\right)$ : Scorer function (inhomogeneous Airy function), $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\in$ : element of, $\int$ : integral, $\mathbb{R}$ : real line, $\sin\NVar{z}$ : sine function and $x$ : real variable
Proof sketch:: Combine (9.5.3) and (9.12.11).
Permalink:: http://dlmf.nist.gov/9.12.E19
Encodings:: pMML, png, TeX
See also:: Annotations for §9.12(vii), §9.12 and Ch.9

►

9.12.20

\operatorname{Hi}\left(z\right)=\frac{1}{\pi}\int_{0}^{\infty}\exp\left(-% \tfrac{1}{3}t^{3}+zt\right)\,\mathrm{d}t,

ⓘ

Symbols:: $\operatorname{Hi}\left(\NVar{z}\right)$ : Scorer function (inhomogeneous Airy function), $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral and $z$ : complex variable
Proof sketch:: Verify by showing that the right-hand side satisfies the differential equation (9.12.1) and the initial conditions given in §9.12(iii).
Referenced by:: (9.12.17), (9.12.31)
Permalink:: http://dlmf.nist.gov/9.12.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §9.12(vii), §9.12 and Ch.9

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