# eta function

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##### 1: 23.16 Graphics
See Figures 23.16.123.16.3 for the modular functions $\lambda$, $J$, and $\eta$. …
##### 2: 23.15 Definitions
###### Dedekind’s EtaFunction (or Dedekind Modular Function)
23.15.9 $\eta\left(\tau\right)=\left(\tfrac{1}{2}\theta_{1}'\left(0,q\right)\right)^{1/% 3}=e^{i\pi\tau/12}\theta_{3}\left(\tfrac{1}{2}\pi(1+\tau)\middle|3\tau\right).$
##### 5: 23.1 Special Notation
The main functions treated in this chapter are the Weierstrass $\wp$-function $\wp\left(z\right)=\wp\left(z|\mathbb{L}\right)=\wp\left(z;g_{2},g_{3}\right)$; the Weierstrass zeta function $\zeta\left(z\right)=\zeta\left(z|\mathbb{L}\right)=\zeta\left(z;g_{2},g_{3}\right)$; the Weierstrass sigma function $\sigma\left(z\right)=\sigma\left(z|\mathbb{L}\right)=\sigma\left(z;g_{2},g_{3}\right)$; the elliptic modular function $\lambda\left(\tau\right)$; Klein’s complete invariant $J\left(\tau\right)$; Dedekind’s eta function $\eta\left(\tau\right)$. …
##### 6: 8.12 Uniform Asymptotic Expansions for Large Parameter
8.12.3 $P\left(a,z\right)=\tfrac{1}{2}\operatorname{erfc}\left(-\eta\sqrt{a/2}\right)-% S(a,\eta),$
8.12.4 $Q\left(a,z\right)=\tfrac{1}{2}\operatorname{erfc}\left(\eta\sqrt{a/2}\right)+S% (a,\eta),$
Then as $a\to\infty$ in the sector $|\operatorname{ph}a|\leq\pi-\delta(<\pi)$,
8.12.7 $S(a,\eta)\sim\frac{e^{-\frac{1}{2}a\eta^{2}}}{\sqrt{2\pi a}}\sum_{k=0}^{\infty% }c_{k}(\eta)a^{-k},$
8.12.8 $T(a,\eta)\sim\frac{e^{\frac{1}{2}a\eta^{2}}}{\sqrt{2\pi a}}\sum_{k=0}^{\infty}% c_{k}(\eta)(-a)^{-k},$
##### 7: 33.11 Asymptotic Expansions for Large $\rho$
###### §33.11 Asymptotic Expansions for Large $\rho$
33.11.1 ${H^{\pm}_{\ell}}\left(\eta,\rho\right)\sim e^{\pm\mathrm{i}{\theta_{\ell}}% \left(\eta,\rho\right)}\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}{\left(b% \right)_{k}}}{k!(\pm 2\mathrm{i}\rho)^{k}},$
33.11.7 $g(\eta,\rho)\widehat{f}(\eta,\rho)-f(\eta,\rho)\widehat{g}(\eta,\rho)=1.$
##### 8: 33.2 Definitions and Basic Properties
###### §33.2(ii) Regular Solution $F_{\ell}\left(\eta,\rho\right)$
$F_{\ell}\left(\eta,\rho\right)$ is a real and analytic function of $\rho$ on the open interval $0<\rho<\infty$, and also an analytic function of $\eta$ when $-\infty<\eta<\infty$. …
###### §33.2(iii) Irregular Solutions $G_{\ell}\left(\eta,\rho\right),{H^{\pm}_{\ell}}\left(\eta,\rho\right)$
Also, $e^{\mp\mathrm{i}{\sigma_{\ell}}\left(\eta\right)}{H^{\pm}_{\ell}}\left(\eta,% \rho\right)$ are analytic functions of $\eta$ when $-\infty<\eta<\infty$.
##### 9: 27.14 Unrestricted Partitions
Dedekind sums occur in the transformation theory of the Dedekind modular function $\eta\left(\tau\right)$, defined by
27.14.12 $\eta\left(\tau\right)=e^{\pi\mathrm{i}\tau/12}\prod_{n=1}^{\infty}(1-e^{2\pi% \mathrm{i}n\tau}),$ $\Im\tau>0$.
$\eta\left(\tau\right)$ satisfies the following functional equation: if $a,b,c,d$ are integers with $ad-bc=1$ and $c>0$, then … For further properties of the function $\eta\left(\tau\right)$ see §§23.1523.19. …
27.14.16 $\Delta\left(\tau\right)=(2\pi)^{12}(\eta\left(\tau\right))^{24},$ $\Im\tau>0$,
##### 10: 23.18 Modular Transformations
###### Dedekind’s EtaFunction
23.18.5 $\eta\left(\mathcal{A}\tau\right)=\varepsilon(\mathcal{A})\left(-i(c\tau+d)% \right)^{1/2}\eta\left(\tau\right),$