# imaginary-modulus transformations

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## 7 matching pages

##### 2: 15.19 Methods of Computation
For $z\in\mathbb{C}$ it is possible to use the linear transformations in such a way that the new arguments lie within the unit circle, except when $z=e^{\pm\pi\mathrm{i}/3}$. This is because the linear transformations map the pair $\{e^{\pi\mathrm{i}/3},e^{-\pi\mathrm{i}/3}\}$ onto itself. …
##### 3: 3.11 Approximation Techniques
Thus if $b_{0}\neq 0$, then the Maclaurin expansion of (3.11.21) agrees with (3.11.20) up to, and including, the term in $z^{p+q}$. …
###### Example. The Discrete Fourier Transform
is called a discrete Fourier transform pair.
##### 4: 20.11 Generalizations and Analogs
If both $m,n$ are positive, then $G(m,n)$ allows inversion of its arguments as a modular transformation (compare (23.15.3) and (23.15.4)):
20.11.2 $\frac{1}{\sqrt{n}}G(m,n)=\frac{1}{\sqrt{n}}\sum\limits_{k=0}^{n-1}e^{-\pi ik^{% 2}m/n}=\frac{e^{-\pi i/4}}{\sqrt{m}}\sum\limits_{j=0}^{m-1}e^{\pi ij^{2}n/m}=% \frac{e^{-\pi i/4}}{\sqrt{m}}G(-n,m).$
With the substitutions $a=qe^{2iz}$, $b=qe^{-2iz}$, with $q=e^{i\pi\tau}$, we have … As in §20.11(ii), the modulus $k$ of elliptic integrals (§19.2(ii)), Jacobian elliptic functions (§22.2), and Weierstrass elliptic functions (§23.6(ii)) can be expanded in $q$-series via (20.9.1). …
##### 5: 10.22 Integrals
where $\psi\left(x\right)=\Gamma'\left(x\right)/\Gamma\left(x\right)$5.2(i)). …
###### §10.22(v) Hankel Transform
The Hankel transform (or Bessel transform) of a function $f(x)$ is defined as … For asymptotic expansions of Hankel transforms see Wong (1976, 1977), Frenzen and Wong (1985a) and Galapon and Martinez (2014). For collections of Hankel transforms see Erdélyi et al. (1954b, Chapter 8) and Oberhettinger (1972). …
##### 6: 15.8 Transformations of Variable
###### §15.8(i) Linear Transformations
15.12.8 $\alpha=\left(-2\ln\left(1-\left(\frac{z-1}{z+1}\right)^{2}\right)\right)^{1/2},$
with the branch chosen to be continuous and $\Re\alpha>0$ when $\Re(\ifrac{(z-1)}{(z+1)})>0$. …
15.12.9 $(z+1)^{3\lambda/2}(2\lambda)^{c-1}\mathbf{F}\left({a+\lambda,b+2\lambda\atop c% };-z\right)={\lambda^{-1/3}\left(e^{\pi\mathrm{i}(a-c+\lambda+(1/3))}\mathrm{% Ai}\left(e^{-\ifrac{2\pi\mathrm{i}}{3}}\lambda^{\ifrac{2}{3}}\beta^{2}\right)+% e^{\pi\mathrm{i}(c-a-\lambda-(1/3))}\mathrm{Ai}\left(e^{\ifrac{2\pi\mathrm{i}}% {3}}\lambda^{\ifrac{2}{3}}\beta^{2}\right)\right)\left(a_{0}(\zeta)+O(\lambda^% {-1})\right)}+\lambda^{-2/3}\left(e^{\pi\mathrm{i}(a-c+\lambda+(2/3))}\mathrm{% Ai}'\left(e^{-\ifrac{2\pi\mathrm{i}}{3}}\lambda^{\ifrac{2}{3}}\beta^{2}\right)% +e^{\pi\mathrm{i}(c-a-\lambda-(2/3))}\mathrm{Ai}'\left(e^{\ifrac{2\pi\mathrm{i% }}{3}}\lambda^{\ifrac{2}{3}}\beta^{2}\right)\right)\left(a_{1}(\zeta)+O(% \lambda^{-1})\right),$
$a_{1}(\zeta)=\left(\tfrac{1}{2}G_{0}(\beta)-\tfrac{1}{2}G_{0}(-\beta)\right)/\beta,$
By combination of the foregoing results of this subsection with the linear transformations of §15.8(i) and the connection formulas of §15.10(ii), similar asymptotic approximations for $F\left(a+e_{1}\lambda,b+e_{2}\lambda;c+e_{3}\lambda;z\right)$ can be obtained with $e_{j}=\pm 1$ or $0$, $j=1,2,3$. …