eta function

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31: 12.17 Physical Applications

… ►By using instead coordinates of the parabolic cylinder $\xi ,\eta ,\zeta$, defined by …

32: Bibliography B

… ►

Coulomb functions in entire ($\eta ,\rho$)-plane. Comput. Phys. Comm. 3 (2), pp. 73–87.

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External Links:

Document, Code

Referenced by:

§33.23(vi).

Encodings:

BibTeX

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Coulomb wave functions for all real $\eta$ and $\rho$ . Comput. Phys. Comm. 8 (5), pp. 377–395.

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Notes:

Single-precision Fortran code for positive energies. For modification see same journal, 11(1976), p. 141.

External Links:

Document, Code

Referenced by:

§3.10(iii), §33.26(ii), §33.8.

Encodings:

BibTeX

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33: 23.21 Physical Applications

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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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34: 22.15 Inverse Functions

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22.15.2 $$\mathrm{cn}(\eta ,k)=x,$$ $-1\le x\le 1$,

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $x$: real, $k$: modulus and $\eta$: solution

Referenced by:

§22.15(i)

Permalink:

http://dlmf.nist.gov/22.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §22.15(i), §22.15 and Ch.22

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22.15.9 $$\eta =\pm \mathrm{arccn}(x,k)+4mK,$$

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{arccn}(x,k)$: inverse Jacobian elliptic function, $x$: real, $k$: modulus, $\eta$: solution and $m$: integer

Permalink:

http://dlmf.nist.gov/22.15.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §22.15(i), §22.15 and Ch.22

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35: Bibliography N

… ►

Coulomb Functions for Large Values of the Parameter $\eta$ . Technical report Atomic Energy of Canada Limited, Chalk River, Ontario.

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Notes:

Rep. CRT-526

External Links:

MathReview (A. Erdélyi)

Referenced by:

§33.7.

Encodings:

BibTeX

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36: 28.32 Mathematical Applications

… ►

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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37: 12.14 The Function $W(a,x)$

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12.14.31 $$W(\frac{1}{2}{\mu }^2,\mu t\sqrt{2})\sim \frac{l(\mu ){\mathrm{e}}^{{\mu }^2\eta }}{2^{\frac{1}{2}}{\mathrm{e}}^{\frac{1}{4}\pi {\mu }^2}{(1-t^2)}^{\frac{1}{4}}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\widetilde{𝒜}}_s(t)}{{\mu }^{2s}},$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $W(a,x)$: parabolic cylinder function, $s$: nonnegative integer, $\eta$ , ${\widetilde{𝒜}}_s(t)$: coefficients and $l(\mu )$

Referenced by:

§12.14(ix)

Permalink:

http://dlmf.nist.gov/12.14.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §12.14(ix), §12.14(ix), §12.14 and Ch.12

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38: 23.22 Methods of Computation

… ►The modular functions $\lambda \left(\tau \right)$, $J\left(\tau \right)$, and $\eta \left(\tau \right)$ are also obtainable in a similar manner from their definitions in §23.15(ii). …

39: 10.41 Asymptotic Expansions for Large Order

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10.41.3 $$I_{\nu }\left(\nu z\right)\sim \frac{{\mathrm{e}}^{\nu \eta }}{{(2\pi \nu )}^{\frac{1}{2}}{(1+z^2)}^{\frac{1}{4}}}\sum _{k=0}^{\mathrm{\infty }}\frac{U_k(p)}{{\nu }^k},$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\eta$ and $U_k(p)$: polynomial coefficient

A&S Ref:

9.7.7

Referenced by:

§10.41(iii), §10.41(iv)

Permalink:

http://dlmf.nist.gov/10.41.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.41(ii), §10.41 and Ch.10

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10.41.4 $$K_{\nu }\left(\nu z\right)\sim {\left(\frac{\pi }{2\nu }\right)}^{\frac{1}{2}}\frac{{\mathrm{e}}^{-\nu \eta }}{{(1+z^2)}^{\frac{1}{4}}}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{U_k(p)}{{\nu }^k},$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\eta$ and $U_k(p)$: polynomial coefficient

A&S Ref:

9.7.8

Permalink:

http://dlmf.nist.gov/10.41.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.41(ii), §10.41 and Ch.10

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10.41.5 $$I_{\nu }^{\prime }\left(\nu z\right)\sim \frac{{(1+z^2)}^{\frac{1}{4}}{\mathrm{e}}^{\nu \eta }}{{(2\pi \nu )}^{\frac{1}{2}}z}\sum _{k=0}^{\mathrm{\infty }}\frac{V_k(p)}{{\nu }^k},$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\eta$ and $V_k(p)$: polynomial coefficient

A&S Ref:

9.7.9

Referenced by:

§10.41(iv)

Permalink:

http://dlmf.nist.gov/10.41.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.41(ii), §10.41 and Ch.10

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10.41.6 $$K_{\nu }^{\prime }\left(\nu z\right)\sim -{\left(\frac{\pi }{2\nu }\right)}^{\frac{1}{2}}\frac{{(1+z^2)}^{\frac{1}{4}}{\mathrm{e}}^{-\nu \eta }}{z}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{V_k(p)}{{\nu }^k},$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\eta$ and $V_k(p)$: polynomial coefficient

A&S Ref:

9.7.10

Referenced by:

§10.41(iii), §10.41(iv)

Permalink:

http://dlmf.nist.gov/10.41.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.41(ii), §10.41 and Ch.10

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10.41.7 $$\eta ={(1+z^2)}^{\frac{1}{2}}+\ln \frac{z}{1+{(1+z^2)}^{\frac{1}{2}}},$$

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Defines:

$\eta$ (locally)

Symbols:

$\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

9.7.11

Permalink:

http://dlmf.nist.gov/10.41.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.41(ii), §10.41 and Ch.10

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40: 28.31 Equations of Whittaker–Hill and Ince

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28.31.3 $$w^{\prime \prime }+\xi \sin \left(2z\right)w^{\prime }+(\eta -p\xi \cos \left(2z\right))w=0.$$

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Symbols:

$\cos z$: cosine function, $\sin z$: sine function, $z$: complex variable, $w(z)$: Mathieu’s equation solution, $\eta$: variable and $\xi$: variable

Referenced by:

§28.31(ii)

Permalink:

http://dlmf.nist.gov/28.31.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.31(ii), §28.31 and Ch.28

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28.31.18 $$w^{\prime \prime }+\left(\eta -\frac{1}{8}{\xi }^2-(p+1)\xi \cos \left(2z\right)+\frac{1}{8}{\xi }^2\cos \left(4z\right)\right)w=0,$$

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Symbols:

$\cos z$: cosine function, $z$: complex variable, $w(z)$: Mathieu’s equation solution and $\eta$: change of variable

Permalink:

http://dlmf.nist.gov/28.31.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §28.31(iii), §28.31 and Ch.28

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