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31: 15.11 Riemann’s Differential Equation

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15.11.2 $$a_1+a_2+b_1+b_2+c_1+c_2=1.$$

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Symbols:

$a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §15.11(i), §15.11 and Ch.15

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15.11.3

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Defines:

: Riemann’s $P$-symbol for solutions of the generalized hypergeometric differential equation

Symbols:

$z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $\alpha$: point, $\beta$: point, $\gamma$: point and $w$: solution

Permalink:

http://dlmf.nist.gov/15.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §15.11(i), §15.11 and Ch.15

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15.11.4

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Symbols:

: Riemann’s $P$-symbol for solutions of the generalized hypergeometric differential equation, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $w$: solution

Permalink:

http://dlmf.nist.gov/15.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §15.11(i), §15.11 and Ch.15

… ►A conformal mapping of the extended complex plane onto itself has the form …where $\kappa$, $\lambda$, $\mu$, $\nu$ are real or complex constants such that $\kappa \nu -\lambda \mu =1$. …

32: 12.1 Special Notation

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$x,y$	real variables.
$z$	complex variable.
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$a,\nu$	real or complex parameters.
…

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33: 15.15 Sums

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15.15.1 $$𝐅(\genfrac{}{}{0pt}{}{a,b}{c};\frac{1}{z})={\left(1-\frac{z_0}{z}\right)}^{-a}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_s}{s!}𝐅(\genfrac{}{}{0pt}{}{-s,b}{c};\frac{1}{z_0}){\left(1-\frac{z}{z_0}\right)}^{-s}.$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $s$: nonnegative integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.19(i)

Permalink:

http://dlmf.nist.gov/15.15.E1

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

The Pochhammer symbol now links to its definition.

See also:

Annotations for §15.15 and Ch.15

►Here $z_0$ ($\ne 0$) is an arbitrary complex constant and the expansion converges when $|z-z_0|>\max (|z_0|,|z_0-1|)$. …

34: 4.3 Graphics

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§4.3(ii) Complex Arguments: Conformal Maps

… ►Corresponding points share the same letters, with bars signifying complex conjugates. … ►

§4.3(iii) Complex Arguments: Surfaces

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Figure 4.3.3: $\ln \left(x+\mathrm{i}y\right)$ (principal value). … Magnify 3D Help

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Figure 4.3.4: ${\mathrm{e}}^{x+\mathrm{i}y}$. Magnify 3D Help

35: 4.29 Graphics

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Figure 4.29.2: Principal values of $\mathrm{arcsinh}x$ and $\mathrm{arccosh}x$. ($\mathrm{arccosh}x$ is complex when .) Magnify

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Figure 4.29.4: Principal values of $\mathrm{arctanh}x$ and $\mathrm{arccoth}x$. ($\mathrm{arctanh}x$ is complex when or $x>1$, and $\mathrm{arccoth}x$ is complex when .) Magnify

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Figure 4.29.6: Principal values of $\mathrm{arccsch}x$ and $\mathrm{arcsech}x$. ($\mathrm{arcsech}x$ is complex when and $x>1$.) Magnify

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§4.29(ii) Complex Arguments

… ►The surfaces for the complex hyperbolic and inverse hyperbolic functions are similar to the surfaces depicted in §4.15(iii) for the trigonometric and inverse trigonometric functions. …

36: 1.1 Special Notation

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$x,y$	real variables.
$z$	complex variable in §§1.2(i), 1.9–1.11, real variable in §§1.5–1.6.
$w$	complex variable in §§1.9–1.11.
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${𝐄}_n$	the space of all $n$-dimensional vectors.
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$\overline{𝐀}$	complex conjugate of the matrix $𝐀$
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►In the physics, applied maths, and engineering literature a common alternative to $\overline{a}$ is $a^{\ast }$, $a$ being a complex number or a matrix; the Hermitian conjugate of $𝐀$ is usually being denoted ${𝐀}^{†}$.

37: 31.11 Expansions in Series of Hypergeometric Functions

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31.11.1 $$w(z)=\sum _{j=0}^{\mathrm{\infty }}{c}_j{P}_j,$$

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Symbols:

$z$: complex variable, $j$: nonnegative integer, $P_j$: General solution of generalized hypergeometric differential equation and $c_j$: coefficients

Referenced by:

§31.11(iii), §31.11(iii), §31.11(iii), §31.11(iii), Erratum (V1.1.7) for Subsection 31.11(iii)

Permalink:

http://dlmf.nist.gov/31.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.11(ii), §31.11 and Ch.31

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31.11.2

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Defines:

$P_j$: General solution of generalized hypergeometric differential equation (locally)

Symbols:

: Riemann’s $P$-symbol for solutions of the generalized hypergeometric differential equation, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $j$: nonnegative integer, $\lambda$ and $\mu$

Referenced by:

§31.11(iii)

Permalink:

http://dlmf.nist.gov/31.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.11(ii), §31.11 and Ch.31

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31.11.3 $$\lambda +\mu =\gamma +\delta -1=\alpha +\beta -ϵ.$$

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Defines:

$\lambda$ (locally) and $\mu$ (locally)

Symbols:

$\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §31.11(ii), §31.11 and Ch.31

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31.11.7 $$L_j=a(\lambda +j)(\mu -j)-q+\frac{(j+\alpha -\mu )(j+\beta -\mu )(j+\gamma -\mu )(j+\lambda )}{(2j+\lambda -\mu )(2j+\lambda -\mu +1)}+\frac{(j-\alpha +\lambda )(j-\beta +\lambda )(j-\gamma +\lambda )(j-\mu )}{(2j+\lambda -\mu )(2j+\lambda -\mu -1)},$$

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Symbols:

$\gamma$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter, $\beta$: real or complex parameter, $\lambda$ , $\mu$ and $L_j$: coefficients

Permalink:

http://dlmf.nist.gov/31.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §31.11(ii), §31.11 and Ch.31

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31.11.12 $$P_j^5=\frac{{\left(\alpha \right)}_j{\left(1-\gamma +\alpha \right)}_j}{{\left(1+\alpha -\beta +ϵ\right)}_{2j}}{z}^{-\alpha -j}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\alpha +j,1-\gamma +\alpha +j}{1+\alpha -\beta +ϵ+2j};\frac{1}{z}),$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $z$: complex variable, $\gamma$: real or complex parameter, $ϵ$: real or complex parameter, $j$: nonnegative integer, $\alpha$: real or complex parameter, $\beta$: real or complex parameter and $P_j^5$: first Kummer solution at $\mathrm{\infty }$ of generalized hypergeometric differential equation

Referenced by:

§31.11(iii), §31.11(iii), §31.11(iii), Erratum (V1.1.7) for Subsection 31.11(iii)

Permalink:

http://dlmf.nist.gov/31.11.E12

Encodings:

pMML, png, TeX

Notational Change (effective with 1.1.7):

We have rewritten the gamma functions in the prefactor more concisely using Pochhammer symbols.

See also:

Annotations for §31.11(iii), §31.11 and Ch.31

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38: 31.9 Orthogonality

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

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

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

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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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31.9.6 $$\rho (s,t)=(s-t){(st)}^{\gamma -1}{\left((s-1)(t-1)\right)}^{\delta -1}{\left((s-a)(t-a)\right)}^{ϵ-1},$$

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Defines:

$\rho (s,t)$: weight function (locally)

Symbols:

$\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter and $a$: complex parameter

Permalink:

http://dlmf.nist.gov/31.9.E6

Encodings:

pMML, png, TeX

See also:

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

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39: 22.8 Addition Theorems

… ►For $u,v\in ℂ$, and with the common modulus $k$ suppressed: ►

22.8.1 $$\mathrm{sn}(u+v)=\frac{\mathrm{sn}u\mathrm{cn}v\mathrm{dn}v+\mathrm{sn}v\mathrm{cn}u\mathrm{dn}u}{1-k^2{\mathrm{sn}}^{2}u{\mathrm{sn}}^{2}v},$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $u$: complex, $v$: complex and $k$: modulus

A&S Ref:

16.17.1

Referenced by:

§22.18(iv)

Permalink:

http://dlmf.nist.gov/22.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §22.8(i), §22.8 and Ch.22

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22.8.2 $$\mathrm{cn}(u+v)=\frac{\mathrm{cn}u\mathrm{cn}v-\mathrm{sn}u\mathrm{dn}u\mathrm{sn}v\mathrm{dn}v}{1-k^2{\mathrm{sn}}^{2}u{\mathrm{sn}}^{2}v},$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $u$: complex, $v$: complex and $k$: modulus

A&S Ref:

16.17.2

Referenced by:

§22.6(ii)

Permalink:

http://dlmf.nist.gov/22.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §22.8(i), §22.8 and Ch.22

… ►For $u,v\in ℂ$, and with the common modulus $k$ suppressed: … ►

22.8.19 $$z_1+z_2+z_3+z_4=0.$$

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Symbols:

$z$: complex

Permalink:

http://dlmf.nist.gov/22.8.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §22.8(iii), §22.8 and Ch.22

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40: 31.2 Differential Equations

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31.2.1 $$\frac{d^{2}w}{{dz}^2}+\left(\frac{\gamma }{z}+\frac{\delta }{z-1}+\frac{ϵ}{z-a}\right)\frac{dw}{dz}+\frac{\alpha \beta z-q}{z(z-1)(z-a)}w=0,$$ $\alpha +\beta +1=\gamma +\delta +ϵ$.

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Referenced by:

§29.2(ii), §31.11(iii), §31.12, §31.13, §31.14(i), §31.17(i), §31.18, §31.2(v), §31.2(v), §31.2(v), §31.2(i), §31.3(i), §31.3(i), §31.3(ii), §31.3(ii), §31.3(ii), §31.3(iii), §31.3(iii), §31.4, §31.5, §31.6, §31.7(ii), §31.8, §31.8, §31.8

Permalink:

http://dlmf.nist.gov/31.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.2(i), §31.2 and Ch.31

… ►All other homogeneous linear differential equations of the second order having four regular singularities in the extended complex plane, $ℂ\cup \{\mathrm{\infty }\}$, can be transformed into (31.2.1). … ►

31.2.2 $$w(z)=z^{-\gamma /2}{(z-1)}^{-\delta /2}{(z-a)}^{-ϵ/2}W(z),$$

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Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $a$: complex parameter and $W(z)$: solution

Permalink:

http://dlmf.nist.gov/31.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.2(ii), §31.2 and Ch.31

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31.2.5 $$z={\sin }^2\theta ,$$

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Symbols:

$\sin z$: sine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/31.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §31.2(iii), §31.2 and Ch.31

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31.2.6 $$\frac{d^{2}w}{{d\theta }^2}+\left((2\gamma -1)\cot \theta -(2\delta -1)\tan \theta -\frac{ϵ\sin \left(2\theta \right)}{a-{\sin }^2\theta }\right)\frac{dw}{d\theta }+4\frac{\alpha \beta {\sin }^2\theta -q}{a-{\sin }^2\theta }w=0.$$

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Symbols:

$\cot z$: cotangent function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\sin z$: sine function, $\tan z$: tangent function, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §31.2(iii), §31.2 and Ch.31

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