克拉斯诺亚尔斯克国立师范大学学历认证办理【假证加微aptao168】mAp

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11: Need Help?

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• How can I view ‘VRML’? What is it? See Viewing DLMF Interactive 3D Graphics.

• What do the colors mean? See About Color Map.

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12: 22.18 Mathematical Applications

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§22.18(ii) Conformal Mapping

►With $k\in [0,1]$ the mapping $z\to w=\mathrm{sn}(z,k)$ gives a conformal map of the closed rectangle $[-K,K]\times [0,K^{\prime }]$ onto the half-plane $\mathrm{\Im }w\ge 0$, with $0,\pm K,\pm K+\mathrm{i}{K}^{\prime },\mathrm{i}{K}^{\prime }$ mapping to $0,\pm 1,\pm k^{-2},\mathrm{\infty }$ respectively. The half-open rectangle $(-K,K)\times [-K^{\prime },K^{\prime }]$ maps onto $ℂ$ cut along the intervals $(-\mathrm{\infty },-1]$ and $[1,\mathrm{\infty })$. See Akhiezer (1990, Chapter 8) and McKean and Moll (1999, Chapter 2) for discussions of the inverse mapping. Bowman (1953, Chapters V–VI) gives an overview of the use of Jacobian elliptic functions in conformal maps for engineering applications. …

13: 23.20 Mathematical Applications

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§23.20(i) Conformal Mappings

… ►The interior of $R$ is mapped one-to-one onto the lower half-plane. … ►For examples of conformal mappings of the function $\mathrm{\wp }\left(z\right)$, see Abramowitz and Stegun (1964, pp. 642–648, 654–655, and 659–60). ►For conformal mappings via modular functions see Apostol (1990, §2.7). …

14: 36.12 Uniform Approximation of Integrals

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36.12.1 $$I(𝐲,k)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp \left(\mathrm{i}kf(u;𝐲)\right)g(u,𝐲)du,$$

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Defines:

$I(𝐲,k)$: oscillatory integral (locally)

Symbols:

$dx$: differential of $x$ , $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, $k$: variable, $g(u,𝐲)$: function, $f(u,𝐲)$: function and $u(t;𝐲)$: mapping

Referenced by:

§36.13

Permalink:

http://dlmf.nist.gov/36.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §36.12(i), §36.12 and Ch.36

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36.12.2 $$\frac{\partial }{\partial u}f(u_j(𝐲);𝐲)=0.$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $f(u,𝐲)$: function and $u(t;𝐲)$: mapping

Permalink:

http://dlmf.nist.gov/36.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §36.12(i), §36.12 and Ch.36

… ►Define a mapping $u(t;𝐲)$ by relating $f(u;𝐲)$ to the normal form (36.2.1) of ${\mathrm{\Phi }}_K(t;𝐱)$ in the following way: ►

36.12.4 $$f(u(t,𝐲);𝐲)=A(𝐲)+{\mathrm{\Phi }}_K(t;𝐱(𝐲)),$$

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Defines:

$u(t;𝐲)$: mapping (locally)

Symbols:

${\mathrm{\Phi }}_K(t;𝐱)$: cuspoid catastrophe of codimension $K$ , $t$: variable, $K$: codimension, $f(u,𝐲)$: function, $𝐱(𝐲)$: function and $A(𝐲)$: function

Permalink:

http://dlmf.nist.gov/36.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §36.12(i), §36.12 and Ch.36

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36.12.6 $$A(𝐲)=f(u(0,𝐲);𝐲),$$

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Defines:

$A(𝐲)$: function (locally)

Symbols:

$f(u,𝐲)$: function and $u(t;𝐲)$: mapping

Permalink:

http://dlmf.nist.gov/36.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §36.12(i), §36.12 and Ch.36

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15: 15.11 Riemann’s Differential Equation

… ►A conformal mapping of the extended complex plane onto itself has the form ►

15.11.5 $$t=(\kappa z+\lambda )/(\mu z+\nu ),$$

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Symbols:

$z$: complex variable, $t$: mapping, $\kappa$: constant, $\lambda$: constant, $\mu$: constant and $\nu$: constant

Permalink:

http://dlmf.nist.gov/15.11.E5

Encodings:

pMML, png, TeX

See also:

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

… ►These constants can be chosen to map any two sets of three distinct points $\{\alpha ,\beta ,\gamma \}$ and $\{\tilde{\alpha },\tilde{\beta },\tilde{\gamma }\}$ onto each other. … ►

15.11.6

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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, $\alpha$: point, $\beta$: point, $\gamma$: point and $t$: mapping

Permalink:

http://dlmf.nist.gov/15.11.E6

Encodings:

pMML, png, TeX

See also:

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

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16: 20.12 Mathematical Applications

… ►The space of complex tori $ℂ/(\mathbb{Z}+\tau \mathbb{Z})$ (that is, the set of complex numbers $z$ in which two of these numbers $z_1$ and $z_2$ are regarded as equivalent if there exist integers $m,n$ such that $z_1-z_2=m+\tau n$) is mapped into the projective space $P^3$ via the identification $z\to ({\theta }_1\left(2z\right|\tau ),{\theta }_2\left(2z\right|\tau ),{\theta }_3\left(2z\right|\tau ),{\theta }_4\left(2z\right|\tau ))$. …

17: 23.16 Graphics

… ►See also About Color Map. …

18: 29.18 Mathematical Applications

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§29.18(iv) Other Applications

►Triebel (1965) gives applications of Lamé functions to the theory of conformal mappings. …

19: 25.8 Sums

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25.8.5 $$\sum _{k=2}^{\mathrm{\infty }}\zeta \left(k\right)z^k=-\gamma z-z\psi \left(1-z\right),$$ .

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Symbols:

$\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\psi \left(z\right)$: psi (or digamma) function, $k$: nonnegative integer and $z$: complex variable

Keywords:

infinite series

Source:

Erdélyi et al. (1953a, (1.17.5), p. 45); with $z\mapsto -z$

Referenced by:

(25.8.7)

Permalink:

http://dlmf.nist.gov/25.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §25.8 and Ch.25

►

25.8.6 $$\sum _{k=0}^{\mathrm{\infty }}\zeta \left(2k\right)z^{2k}=-\frac{1}{2}\pi z\cot \left(\pi z\right),$$ .

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $k$: nonnegative integer and $z$: complex variable

Keywords:

infinite series

Source:

Erdélyi et al. (1953a, (1.20.3), p. 51); with $z\mapsto \pi z$

Referenced by:

(25.8.8)

Permalink:

http://dlmf.nist.gov/25.8.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §25.8 and Ch.25

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20: 5.3 Graphics

… ►See also About Color Map. …