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21: 28.25 Asymptotic Expansions for Large $\mathrm{\Re }z$

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28.25.1 $${\mathrm{M}}_{\nu }^{(3,4)}(z,h)\sim \frac{{\mathrm{e}}^{\pm \mathrm{i}\left(2h\cosh z-\left(\frac{1}{2}\nu +\frac{1}{4}\right)\pi \right)}}{{\left(\pi h(\cosh z+1)\right)}^{\frac{1}{2}}}\sum _{m=0}^{\mathrm{\infty }}\frac{D_m^{\pm }}{{\left(\mp 4\mathrm{i}h(\cosh z+1)\right)}^m},$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable, $\nu$: complex parameter and $D_m^{\pm }$: coefficients

A&S Ref:

20.9.1 (for 3 and for integer $\nu$) 20.9.3 (for 4 and for integer $\nu$)

Referenced by:

§28.25

Permalink:

http://dlmf.nist.gov/28.25.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.25 and Ch.28

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28.25.3 $$(m+1)D_{m+1}^{\pm }+\left({(m+\frac{1}{2})}^2\pm (m+\frac{1}{4})8\mathrm{i}h+2h^2-a\right)D_m^{\pm }\pm (m-\frac{1}{2})\left(8\mathrm{i}hm\right)D_{m-1}^{\pm }=0,$$ $m\ge 0$.

ⓘ

Symbols:

$\mathrm{i}$: imaginary unit, $m$: integer, $h$: parameter, $a$: parameter and $D_m^{\pm }$: coefficients

A&S Ref:

20.9.2 (for $+$) 20.9.4 (for $-$)

Permalink:

http://dlmf.nist.gov/28.25.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.25 and Ch.28

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22: 28.5 Second Solutions ${\mathrm{fe}}_n$, ${\mathrm{ge}}_n$

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28.5.1 $${\mathrm{fe}}_n(z,q)=C_n(q)\left(z{\mathrm{ce}}_n(z,q)+f_n(z,q)\right),$$

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Defines:

${\mathrm{fe}}_n(z,q)$: second solution, Mathieu’s equation

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $q=h^2$: parameter, $n$: integer, $z$: complex variable, $f_n(z,q)$: functions and $C_n(q)$: factor

A&S Ref:

20.3.6 (in different form)

Referenced by:

§28.5(i), §28.5(i)

Permalink:

http://dlmf.nist.gov/28.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.5(i), §28.5(i), §28.5 and Ch.28

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28.5.2 $${\mathrm{ge}}_n(z,q)=S_n(q)\left(z{\mathrm{se}}_n(z,q)+g_n(z,q)\right),$$

ⓘ

Defines:

${\mathrm{ge}}_n(z,q)$: second solution, Mathieu’s equation

Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $q=h^2$: parameter, $n$: integer, $z$: complex variable, $g_n(z,q)$: functions and $S_n(q)$: factor

A&S Ref:

20.3.7 (in different form)

Referenced by:

§28.5(i), §28.5(i)

Permalink:

http://dlmf.nist.gov/28.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.5(i), §28.5(i), §28.5 and Ch.28

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28.5.3

ⓘ

Defines:

$f_n(z,q)$: functions (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $q=h^2$: parameter, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/28.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.5(i), §28.5(i), §28.5 and Ch.28

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28.5.4

ⓘ

Defines:

$g_n(z,q)$: functions (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $m$: integer, $q=h^2$: parameter, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/28.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.5(i), §28.5(i), §28.5 and Ch.28

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28.5.5 $${(C_n(q))}^2{\int }_0^{2\pi }{(f_n(x,q))}^{2}dx={(S_n(q))}^2{\int }_0^{2\pi }{(g_n(x,q))}^{2}dx=\pi .$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $q=h^2$: parameter, $n$: integer, $x$: real variable, $f_n(z,q)$: functions, $g_n(z,q)$: functions, $C_n(q)$: factor and $S_n(q)$: factor

Permalink:

http://dlmf.nist.gov/28.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.5(i), §28.5(i), §28.5 and Ch.28

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23: 31.16 Mathematical Applications

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31.16.1 $${\mathrm{𝐻𝑝}}_{n,m}\left(x\right){\mathrm{𝐻𝑝}}_{n,m}\left(y\right)=\sum _{j=0}^n{A}_j{\sin }^{2j}\theta P_{n-j}^{(\gamma +\delta +2j-1,ϵ-1)}\left(\cos \left(2\theta \right)\right)P_j^{(\delta -1,\gamma -1)}\left(\cos \left(2\varphi \right)\right),$$

ⓘ

Symbols:

${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$: Heun polynomials, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\cos z$: cosine function, $\sin z$: sine function, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $j$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\beta$: real or complex parameter, $\theta$: angle, $\varphi$: angle, $x$: variable, $y$: variable and $A_j$: coefficients

Referenced by:

§31.16(ii), §31.16(ii)

Permalink:

http://dlmf.nist.gov/31.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.16(ii), §31.16 and Ch.31

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31.16.3 $$A_0=\frac{n!}{{\left(\gamma +\delta \right)}_n}{\mathrm{𝐻𝑝}}_{n,m}\left(1\right),Q_0{A}_0+R_0{A}_1=0,$$

ⓘ

Symbols:

${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$: Heun polynomials, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $m$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\beta$: real or complex parameter, $A_j$: coefficients, $Q_j$ and $R_j$

Referenced by:

§31.16(ii), Erratum (V1.0.21) for Equations (31.16.2) and (31.16.3), Erratum (V1.0.21) for Equations (31.16.2) and (31.16.3)

Permalink:

http://dlmf.nist.gov/31.16.E3

Encodings:

pMML, png, TeX

Correction (effective with 1.0.21):

The initial data $A_0$, previously missing, has now been included.

See also:

Annotations for §31.16(ii), §31.16 and Ch.31

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31.16.5 $$P_j=\frac{(ϵ-j+n)j(\beta +j-1)(\gamma +\delta +j-2)}{(\gamma +\delta +2j-3)(\gamma +\delta +2j-2)},$$

ⓘ

Defines:

$P_j$ (locally)

Symbols:

$\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.16.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §31.16(ii), §31.16 and Ch.31

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31.16.6 $$Q_j=-aj(j+\gamma +\delta -1)-q+\frac{(j-n)(j+\beta )(j+\gamma )(j+\gamma +\delta -1)}{(2j+\gamma +\delta )(2j+\gamma +\delta -1)}+\frac{(j+n+\gamma +\delta -1)j(j+\delta -1)(j-\beta +\gamma +\delta -1)}{(2j+\gamma +\delta -1)(2j+\gamma +\delta -2)},$$

ⓘ

Defines:

$Q_j$ (locally)

Symbols:

$\gamma$: real or complex parameter, $\delta$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.16.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §31.16(ii), §31.16 and Ch.31

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31.16.7 $$R_j=\frac{(n-j)(j+n+\gamma +\delta )(j+\gamma )(j+\delta )}{(\gamma +\delta +2j)(\gamma +\delta +2j+1)}.$$

ⓘ

Defines:

$R_j$ (locally)

Symbols:

$\gamma$: real or complex parameter, $\delta$: real or complex parameter, $j$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/31.16.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §31.16(ii), §31.16 and Ch.31

…

24: 31.6 Path-Multiplicative Solutions

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31.6.1 $$(s_1,s_2){\mathrm{𝐻𝑓}}_m^{\nu }(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z),$$ $m=0,1,2,\mathrm{\dots }$,

ⓘ

Symbols:

$(s_1,s_2){\mathrm{𝐻𝑓}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $\nu$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter, $\beta$: real or complex parameter and $s_j$: parameters

Permalink:

http://dlmf.nist.gov/31.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.6 and Ch.31

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25: 31.9 Orthogonality

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31.9.1 $$w_m(z)=(0,1){\mathrm{𝐻𝑓}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z),$$

ⓘ

Symbols:

$(s_1,s_2){\mathrm{𝐻𝑓}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.9(i), §31.9 and Ch.31

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31.9.2 $${\int }_{\zeta }^{(1+,0+,1-,0-)}{t}^{\gamma -1}{(1-t)}^{\delta -1}{(t-a)}^{ϵ-1}{w}_m(t)w_k(t)dt={\delta }_{m,k}{\theta }_m.$$

ⓘ

Symbols:

${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\int$: integral, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $\zeta$: point and ${\theta }_m$: normalization constant

Permalink:

http://dlmf.nist.gov/31.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.9(i), §31.9 and Ch.31

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31.9.3 $${\theta }_m={(1-{\mathrm{e}}^{2\pi \mathrm{i}\gamma })(1-{\mathrm{e}}^{2\pi \mathrm{i}\delta }){\zeta }^{\gamma }{(1-\zeta )}^{\delta }{(\zeta -a)}^{ϵ}\frac{f_0(q,\zeta )}{f_1(q,\zeta )}\frac{\partial }{\partial q}𝒲\left\{f_0(q,\zeta ),f_1(q,\zeta )\right\}|}_{q=q_m},$$

ⓘ

Defines:

${\theta }_m$: normalization constant (locally)

Symbols:

$𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\zeta$: point and $f_0(q_m,z)$, $f_1(q_m,z)$: coefficients

Permalink:

http://dlmf.nist.gov/31.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §31.9(i), §31.9 and Ch.31

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26: 31.4 Solutions Analytic at Two Singularities: Heun Functions

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31.4.1 $$(0,1){\mathrm{𝐻𝑓}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z),$$ $m=0,1,2,\mathrm{\dots }$.

ⓘ

Symbols:

$(s_1,s_2){\mathrm{𝐻𝑓}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Referenced by:

§31.6

Permalink:

http://dlmf.nist.gov/31.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.4 and Ch.31

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31.4.3 $$(s_1,s_2){\mathrm{𝐻𝑓}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z),$$ $m=0,1,2,\mathrm{\dots }$,

ⓘ

Symbols:

$(s_1,s_2){\mathrm{𝐻𝑓}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Referenced by:

§31.4, §31.6

Permalink:

http://dlmf.nist.gov/31.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §31.4 and Ch.31

…

27: 33.19 Power-Series Expansions in $r$

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33.19.1 $$f(ϵ,\mathrm{\ell };r)=r^{\mathrm{\ell }+1}\sum _{k=0}^{\mathrm{\infty }}{\alpha }_k{r}^k,$$

ⓘ

Symbols:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter and ${\alpha }_k$: term

Referenced by:

§33.18, §33.19

Permalink:

http://dlmf.nist.gov/33.19.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.19 and Ch.33

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33.19.3 $$2\pi h(ϵ,\mathrm{\ell };r)=\sum _{k=0}^{2\mathrm{\ell }}\frac{(2\mathrm{\ell }-k)!{\gamma }_k}{k!}{(2r)}^{k-\mathrm{\ell }}-\sum _{k=0}^{\mathrm{\infty }}{\delta }_k{r}^{k+\mathrm{\ell }+1}-A(ϵ,\mathrm{\ell })\left(2\ln |2r/\kappa |+\mathrm{\Re }\psi \left(\mathrm{\ell }+1+\kappa \right)+\mathrm{\Re }\psi \left(-\mathrm{\ell }+\kappa \right)\right)f(ϵ,\mathrm{\ell };r),$$ $r\ne 0$.

ⓘ

Symbols:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $h(ϵ,\mathrm{\ell };r)$: irregular Coulomb function, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter, $\kappa$: quantity, $A(ϵ,\mathrm{\ell })$: function, ${\gamma }_k$: term and ${\delta }_k$: term

Referenced by:

§33.19

Permalink:

http://dlmf.nist.gov/33.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.19 and Ch.33

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33.19.4 $${\gamma }_k-{\gamma }_{k-1}+\frac{1}{4}(k-1)(k-2\mathrm{\ell }-2)ϵ{\gamma }_{k-2}=0,$$ $k=2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $ϵ$: real parameter and ${\gamma }_k$: term

Permalink:

http://dlmf.nist.gov/33.19.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §33.19 and Ch.33

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33.19.6 $$k(k+2\mathrm{\ell }+1){\delta }_k+2{\delta }_{k-1}+ϵ{\delta }_{k-2}+2(2k+2\mathrm{\ell }+1)A(ϵ,\mathrm{\ell }){\alpha }_k=0,$$ $k=2,3,\mathrm{\dots }$,

ⓘ

Symbols:

$k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $ϵ$: real parameter, $A(ϵ,\mathrm{\ell })$: function, ${\alpha }_k$: term and ${\delta }_k$: term

Permalink:

http://dlmf.nist.gov/33.19.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §33.19 and Ch.33

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33.19.7 $${\beta }_k-{\beta }_{k-1}+\frac{1}{4}(k-1)(k-2\mathrm{\ell }-2)ϵ{\beta }_{k-2}+\frac{1}{2}(k-1)ϵ{\gamma }_{k-2}=0,$$ $k=2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $ϵ$: real parameter, ${\gamma }_k$: term and ${\beta }_k$: term

Permalink:

http://dlmf.nist.gov/33.19.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §33.19 and Ch.33

…

28: 28.20 Definitions and Basic Properties

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28.20.6 $${\mathrm{Fe}}_n(z,q)=\mp \mathrm{i}{\mathrm{fe}}_n(\pm \mathrm{i}z,q),$$ $n=0,1,\mathrm{\dots }$,

ⓘ

Defines:

${\mathrm{Fe}}_n(z,q)$: modified Mathieu function

Symbols:

${\mathrm{fe}}_n(z,q)$: second solution, Mathieu’s equation, $\mathrm{i}$: imaginary unit, $q=h^2$: parameter, $n$: integer and $z$: complex variable

A&S Ref:

20.6.1 (in slightly different form) 20.6.2 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.20.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(ii), §28.20 and Ch.28

►

28.20.7 $${\mathrm{Ge}}_n(z,q)={\mathrm{ge}}_n(\pm \mathrm{i}z,q),$$ $n=1,2,\mathrm{\dots }$.

ⓘ

Defines:

${\mathrm{Ge}}_n(z,q)$: modified Mathieu function

Symbols:

${\mathrm{ge}}_n(z,q)$: second solution, Mathieu’s equation, $\mathrm{i}$: imaginary unit, $q=h^2$: parameter, $n$: integer and $z$: complex variable

A&S Ref:

20.6.1 (in slightly different form) 20.6.2 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.20.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(ii), §28.20 and Ch.28

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28.20.17 $${\mathrm{Ie}}_n(z,h)={\mathrm{i}}^{-n}{\mathrm{Mc}}_n^{(1)}(z,\mathrm{i}h),$$

ⓘ

Defines:

${\mathrm{Ie}}_n(z,h)$: modified Mathieu function

Symbols:

$\mathrm{i}$: imaginary unit, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $h$: parameter, $n$: integer and $z$: complex variable

A&S Ref:

20.8.10 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.20.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(v), §28.20 and Ch.28

►

28.20.18 $${\mathrm{Io}}_n(z,h)={\mathrm{i}}^{-n}{\mathrm{Ms}}_n^{(1)}(z,\mathrm{i}h),$$

ⓘ

Defines:

${\mathrm{Io}}_n(z,h)$: modified Mathieu function

Symbols:

$\mathrm{i}$: imaginary unit, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $h$: parameter, $n$: integer and $z$: complex variable

A&S Ref:

20.8.10 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.20.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(v), §28.20 and Ch.28

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28.20.22 $${\mathrm{M}}_{\nu }^{(j)}(z\pm \frac{1}{2}\pi \mathrm{i},h)={\mathrm{M}}_{\nu }^{(j)}(z,\pm \mathrm{i}h),$$ $\nu \notin \mathbb{Z}$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $\notin$: not an element of, $h$: parameter, $j$: integer, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.20.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(vii), §28.20 and Ch.28

…

29: 13.2 Definitions and Basic Properties

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Kummer’s Equation

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13.2.11 $$U(a,-n,z)=z^{n+1}U(a+n+1,n+2,z).$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

…

30: 36.8 Convergent Series Expansions

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36.8.3 $$\frac{3^{2/3}}{4{\pi }^2}{\mathrm{\Psi }}^{(\mathrm{H})}\left(3^{1/3}𝐱\right)=\mathrm{Ai}\left(x\right)\mathrm{Ai}\left(y\right)\sum _{n=0}^{\mathrm{\infty }}{(-3^{-1/3}\mathrm{i}z)}^n\frac{c_n(x)c_n(y)}{n!}+\mathrm{Ai}\left(x\right){\mathrm{Ai}}^{\prime }\left(y\right)\sum _{n=2}^{\mathrm{\infty }}{(-3^{-1/3}\mathrm{i}z)}^n\frac{c_n(x)d_n(y)}{n!}+{\mathrm{Ai}}^{\prime }\left(x\right)\mathrm{Ai}\left(y\right)\sum _{n=2}^{\mathrm{\infty }}{(-3^{-1/3}\mathrm{i}z)}^n\frac{d_n(x)c_n(y)}{n!}+{\mathrm{Ai}}^{\prime }\left(x\right){\mathrm{Ai}}^{\prime }\left(y\right)\sum _{n=1}^{\mathrm{\infty }}{(-3^{-1/3}\mathrm{i}z)}^n\frac{d_n(x)d_n(y)}{n!},$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $y$: real parameter, $z$: real parameter, $m$: integer, $x$: real parameter, $c_n(t)$: coefficients and $d_n(t)$: coefficients

Referenced by:

§36.8

Permalink:

http://dlmf.nist.gov/36.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §36.8 and Ch.36

… ►

36.8.4 $${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)=2{\pi }^2{\left(\frac{2}{3}\right)}^{2/3}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(-\mathrm{i}{(2/3)}^{2/3}z\right)}^n}{n!}\mathrm{\Re }\left(f_n(\frac{x+\mathrm{i}y}{{12}^{1/3}},\frac{x-\mathrm{i}y}{{12}^{1/3}})\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\mathrm{\Re }$: real part, $y$: real parameter, $z$: real parameter, $m$: integer, $x$: real parameter and $f_n$: function

Referenced by:

§36.8

Permalink:

http://dlmf.nist.gov/36.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §36.8 and Ch.36

…