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21: 1.9 Calculus of a Complex Variable

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${\displaystyle \frac{\partial u}{\partial x}}={\displaystyle \frac{\partial v}{\partial y}},$

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${\displaystyle \frac{\partial u}{\partial y}}=-{\displaystyle \frac{\partial v}{\partial x}}$

… ►Conversely, if at a given point $(x,y)$ the partial derivatives $\partial u/\partial x$, $\partial u/\partial y$, $\partial v/\partial x$, and $\partial v/\partial y$ exist, are continuous, and satisfy (1.9.25), then $f(z)$ is differentiable at $z=x+\mathrm{i}y$. … ►

Bilinear Transformation

… ►Other names for the bilinear transformation are fractional linear transformation, homographic transformation, and Möbius transformation. …

22: 21.7 Riemann Surfaces

… ►by setting $\lambda =\tilde{\lambda }/\tilde{\eta }$, $\mu =\tilde{\mu }/\tilde{\eta }$, and then clearing fractions. … ►

21.7.7 $$({\frac{\partial }{\partial z_1}\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)|}_{𝐳=\mathrm{𝟎}},\mathrm{\dots },{\frac{\partial }{\partial z_g}\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)|}_{𝐳=\mathrm{𝟎}})\ne \mathrm{𝟎}.$$

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Symbols:

$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $g$: positive integer, $𝛀$: a Riemann matrix, $𝜶$: $g$-dimensional vector and $𝜷$: $g$-dimensional vector

Permalink:

http://dlmf.nist.gov/21.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §21.7(ii), §21.7 and Ch.21

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21.7.8 $$\zeta ={\sum _{j=1}^g{\omega }_j\frac{\partial }{\partial z_j}\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)|}_{𝐳=\mathrm{𝟎}}.$$

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Symbols:

$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $g$: positive integer, $𝛀$: a Riemann matrix, $𝜶$: $g$-dimensional vector, $𝜷$: $g$-dimensional vector and $\omega$: differential

Permalink:

http://dlmf.nist.gov/21.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §21.7(ii), §21.7 and Ch.21

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23: 9.18 Tables

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Zhang and Jin (1996, p. 337) tabulates $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$, $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$ for $x=0(1)20$ to 8S and for $x=-20(1)0$ to 9D.

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Miller (1946) tabulates $a_k$, ${\mathrm{Ai}}^{\prime }\left(a_k\right)$, $a_k^{\prime }$, $\mathrm{Ai}\left(a_k^{\prime }\right)$, $k=1(1)50$; $b_k$, ${\mathrm{Bi}}^{\prime }\left(b_k\right)$, $b_k^{\prime }$, $\mathrm{Bi}\left(b_k^{\prime }\right)$, $k=1(1)20$. Precision is 8D. Entries for $k=1(1)20$ are reproduced in Abramowitz and Stegun (1964, Chapter 10).

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Sherry (1959) tabulates $a_k$, ${\mathrm{Ai}}^{\prime }\left(a_k\right)$, $a_k^{\prime }$, $\mathrm{Ai}\left(a_k^{\prime }\right)$, $k=1(1)50$; 20S.

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Zhang and Jin (1996, p. 339) tabulates $a_k$, ${\mathrm{Ai}}^{\prime }\left(a_k\right)$, $a_k^{\prime }$, $\mathrm{Ai}\left(a_k^{\prime }\right)$, $b_k$, ${\mathrm{Bi}}^{\prime }\left(b_k\right)$, $b_k^{\prime }$, $\mathrm{Bi}\left(b_k^{\prime }\right)$, $k=1(1)20$; 8D.

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Smirnov (1960) tabulates $U_1(x,\alpha )$, $U_2(x,\alpha )$, defined by (9.13.20), (9.13.21), and also $\partial U_1(x,\alpha )/\partial x$, $\partial U_2(x,\alpha )/\partial x$, for $\alpha =1$, $x=-6(.01)10$ to 5D or 5S, and also for $\alpha =\pm \frac{1}{4}$, $\pm \frac{1}{3}$, $\pm \frac{1}{2}$, $\pm \frac{2}{3}$, $\pm \frac{3}{4}$, $\frac{5}{4}$, $\frac{4}{3}$, $\frac{3}{2}$, $\frac{5}{3}$, $\frac{7}{4}$, 2, $x=0(.01)6$; 4D.

24: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

§22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

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22.12.13 $$2K\mathrm{cs}(2Kt,k)=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\frac{\pi }{\tan \left(\pi (t-n\tau )\right)}=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\left(\underset{M\to \mathrm{\infty }}{lim}\sum _{m=-M}^M\frac{1}{t-m-n\tau }\right).$$

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Symbols:

$\mathrm{cs}(z,k)$: 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, $\tan z$: tangent function, $k$: modulus and $\tau$: ratio

Referenced by:

§22.12

Permalink:

http://dlmf.nist.gov/22.12.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

25: 33.23 Methods of Computation

… ►Cancellation errors increase with increases in $\rho$ and $|r|$, and may be estimated by comparing the final sum of the series with the largest partial sum. … ►

§33.23(v) Continued Fractions

►§33.8 supplies continued fractions for $F_{\mathrm{\ell }}^{\prime }/F_{\mathrm{\ell }}$ and $H_{\mathrm{\ell }}^{\pm }{}_{}{}^{\prime }/H_{\mathrm{\ell }}^{\pm }$. … ►Thompson and Barnett (1985, 1986) and Thompson (2004) use combinations of series, continued fractions, and Padé-accelerated asymptotic expansions (§3.11(iv)) for the analytic continuations of Coulomb functions. …

26: 1.10 Functions of a Complex Variable

… ►Let $D$ be a bounded domain with boundary $\partial D$ and let $\overline{D}=D\cup \partial D$. If $f(z)$ is continuous on $\overline{D}$ and analytic in $D$, then $\left|f(z)\right|$ attains its maximum on $\partial D$. … ►The convergence of the infinite product is uniform if the sequence of partial products converges uniformly. … ►

§1.10(x) Infinite Partial Fractions

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Mittag-Leffler’s Expansion

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27: 23.21 Physical Applications

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§23.21(ii) Nonlinear Evolution Equations

►Airault et al. (1977) applies the function $\mathrm{\wp }$ to an integrable classical many-body problem, and relates the solutions to nonlinear partial differential equations. … ►

23.21.2 $$(\eta -\zeta )(\zeta -\xi )(\xi -\eta ){\nabla }^2=(\zeta -\eta )f(\xi )f^{\prime }(\xi )\frac{\partial }{\partial \xi }+(\xi -\zeta )f(\eta )f^{\prime }(\eta )\frac{\partial }{\partial \eta }+(\eta -\xi )f(\zeta )f^{\prime }(\zeta )\frac{\partial }{\partial \zeta },$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\xi$: ellipsoidal coordinate, $\eta$: ellipsoidal coordinate, $\zeta$: ellipsoidal coordinate and $f(\rho )$: function

Permalink:

http://dlmf.nist.gov/23.21.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §23.21(iii), §23.21 and Ch.23

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23.21.5 $$\left(\mathrm{\wp }\left(v\right)-\mathrm{\wp }\left(w\right)\right)\left(\mathrm{\wp }\left(w\right)-\mathrm{\wp }\left(u\right)\right)\left(\mathrm{\wp }\left(u\right)-\mathrm{\wp }\left(v\right)\right){\nabla }^2=\left(\mathrm{\wp }\left(w\right)-\mathrm{\wp }\left(v\right)\right)\frac{{\partial }^2}{{\partial u}^2}+\left(\mathrm{\wp }\left(u\right)-\mathrm{\wp }\left(w\right)\right)\frac{{\partial }^2}{{\partial v}^2}+\left(\mathrm{\wp }\left(v\right)-\mathrm{\wp }\left(u\right)\right)\frac{{\partial }^2}{{\partial w}^2}.$$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $𝕃$: lattice, $u$: change of variable, $v$: change of variable and $w$: change of variable

Permalink:

http://dlmf.nist.gov/23.21.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §23.21(iii), §23.21 and Ch.23

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28: 3.8 Nonlinear Equations

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3.8.13 $$\frac{dz}{d\alpha }=-\frac{\partial f}{\partial \alpha }/\frac{\partial f}{\partial z}.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $\alpha$: parameter

Permalink:

http://dlmf.nist.gov/3.8.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §3.8(vi), §3.8 and Ch.3

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3.8.15 $$p(x)=(x-1)(x-2)\mathrm{\cdots }(x-20)$$

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Permalink:

http://dlmf.nist.gov/3.8.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §3.8(vi), §3.8(vi), §3.8 and Ch.3

… ►Consider $x=20$ and $j=19$. We have $p^{\prime }(20)=19!$ and $a_{19}=1+2+\mathrm{\cdots }+20=210$. … ►

3.8.16 $$\frac{dx}{d{a}_{19}}=-\frac{{20}^{19}}{19!}=(-4.30\mathrm{\dots })\times {10}^7.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/3.8.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §3.8(vi), §3.8(vi), §3.8 and Ch.3

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29: 28.32 Mathematical Applications

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28.32.2 $$\frac{{\partial }^{2}V}{{\partial x}^2}+\frac{{\partial }^{2}V}{{\partial y}^2}+k^{2}V=0$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $x$: real variable, $y$: real variable, $k$: parameter and $V(\xi ,\eta )$: solution

Referenced by:

§28.33(ii)

Permalink:

http://dlmf.nist.gov/28.32.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.32(i), §28.32 and Ch.28

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28.32.3 $$\frac{{\partial }^{2}V}{{\partial \xi }^2}+\frac{{\partial }^{2}V}{{\partial \eta }^2}+\frac{1}{2}{c}^2{k}^2(\cosh \left(2\xi \right)-\cos \left(2\eta \right))V=0.$$

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Symbols:

$\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\eta$: variable, $\xi$: variable, $k$: parameter and $V(\xi ,\eta )$: solution

Referenced by:

§28.32(i), §28.33(ii)

Permalink:

http://dlmf.nist.gov/28.32.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.32(i), §28.32 and Ch.28

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28.32.4 $$\frac{{\partial }^{2}K}{{\partial z}^2}-\frac{{\partial }^{2}K}{{\partial \zeta }^2}=2q\left(\cos \left(2z\right)-\cos \left(2\zeta \right)\right)K.$$

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Symbols:

$\cos z$: cosine function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $q=h^2$: parameter, $z$: complex variable, $\zeta$: variable and $K(z,\zeta )$: solution

Permalink:

http://dlmf.nist.gov/28.32.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.32(i), §28.32 and Ch.28

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28.32.5 $$K(z,\zeta )\frac{du(\zeta )}{d\zeta }-u(\zeta )\frac{\partial K(z,\zeta )}{\partial \zeta }$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $z$: complex variable, $\zeta$: variable, $u(\zeta )$: solution and $K(z,\zeta )$: solution

Permalink:

http://dlmf.nist.gov/28.32.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.32(i), §28.32 and Ch.28

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30: 20.13 Physical Applications

… ►The functions ${\theta }_j\left(z\right|\tau )$, $j=1,2,3,4$, provide periodic solutions of the partial differential equation ►

20.13.1 $$\partial \theta (z|\tau )/\partial \tau =\kappa {\partial }^2\theta (z|\tau )/{\partial z}^2,$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\tau$: lattice parameter and $z$: real variable

Referenced by:

§20.13

Permalink:

http://dlmf.nist.gov/20.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §20.13 and Ch.20

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20.13.2 $$\partial \theta /\partial t=\alpha {\partial }^2\theta /{\partial z}^2,$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $t$: time, $\alpha$: real parameter and $z$: real variable

Referenced by:

§20.13

Permalink:

http://dlmf.nist.gov/20.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §20.13 and Ch.20

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