change of amplitude

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6 matching pages

1: 19.7 Connection Formulas

… ►

§19.7(ii) Change of Modulus and Amplitude

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2: 29.2 Differential Equations

… ►

29.2.5

\phi=\tfrac{1}{2}\pi-\operatorname{am}\left(z,k\right).

ⓘ

Defines:: $\phi$ : change of variable (locally)
Symbols:: $\operatorname{am}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s amplitude function, $\pi$ : the ratio of the circumference of a circle to its diameter, $z$ : complex variable and $k$ : real parameter
Referenced by:: §29.15(i), §29.15(ii), §29.6(i)
Permalink:: http://dlmf.nist.gov/29.2.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §29.2(ii), §29.2 and Ch.29

…

3: 29.1 Special Notation

… ►where

\psi=\operatorname{am}\left(z,k\right)

; see §22.16(i). The relation to the Lamé functions

{\rm Ec}^{m}_{\nu}

{\rm Es}^{m}_{\nu}

of Ince (1940b) is given by …

4: 29.6 Fourier Series

… ►With

\phi=\frac{1}{2}\pi-\operatorname{am}\left(z,k\right)

, as in (29.2.5), we have …

5: Errata

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Changes

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Changes

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Changes

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Changes

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Changes

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6: 19.25 Relations to Other Functions

… ►then the five nontrivial permutations of

x, y, z

that leave

R_{F}

invariant change

k^{2}

(

=(z-y)/(z-x)

) into

1/k^{2}

{k^{\prime}}^{2}

1/{k^{\prime}}^{2}

-k^{2}/{k^{\prime}}^{2}

-{k^{\prime}}^{2}/k^{2}

, and

\sin\phi

(

=\sqrt{(z-x)/z}

) into

k\sin\phi

-i\tan\phi

-ik^{\prime}\tan\phi

(k^{\prime}\sin\phi)/\sqrt{1-k^{2}{\sin}^{2}\phi}

-ik\sin\phi/\sqrt{1-k^{2}{\sin}^{2}\phi}

. … ►The three changes of parameter of

\Pi\left(\phi,\alpha^{2},k\right)

in §19.7(iii) are unified in (19.21.12) by way of (19.25.14). … ►

19.25.30

\operatorname{am}\left(u,k\right)=R_{C}\left({\operatorname{cs}}^{2}\left(u,k% \right),{\operatorname{ns}}^{2}\left(u,k\right)\right),

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Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\operatorname{am}\left(\NVar{x},\NVar{k}\right)$ : Jacobi’s amplitude function, $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function and $k$ : real or complex modulus
Referenced by:: §19.25(v)
Permalink:: http://dlmf.nist.gov/19.25.E30
Encodings:: pMML, png, TeX
See also:: Annotations for §19.25(v), §19.25 and Ch.19

… ►The sign on the right-hand side of (19.25.35) will change whenever one crosses a curve on which

\wp\left(z\right)-e_{j}<0

, for some

j

. … ►The sign on the right-hand side of (19.25.40) will change whenever one crosses a curve on which

\sigma_{j}^{2}(z)<0

, for some

j

. …