# change of modulus

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## 11—20 of 84 matching pages

##### 11: 1.13 Differential Equations
The substitution $\xi=1/z$ in (1.13.1) gives …
1.13.21 $\left\{z,\zeta\right\}=(\ifrac{\mathrm{d}\xi}{\mathrm{d}\zeta})^{2}\left\{z,% \xi\right\}+\left\{\xi,\zeta\right\}.$
##### 12: 15.8 Transformations of Variable
With $\zeta=e^{\ifrac{2\pi\mathrm{i}}{3}}(1-z)/\left(z-e^{\ifrac{4\pi\mathrm{i}}{3}}\right)$
##### 13: 20.11 Generalizations and Analogs
20.11.2 $\frac{1}{\sqrt{n}}G(m,n)=\frac{1}{\sqrt{n}}\sum\limits_{k=0}^{n-1}e^{-\pi ik^{% 2}m/n}=\frac{e^{-\pi i/4}}{\sqrt{m}}\sum\limits_{j=0}^{m-1}e^{\pi ij^{2}n/m}=% \frac{e^{-\pi i/4}}{\sqrt{m}}G(-n,m).$
20.11.3 $f(a,b)=\sum_{n=-\infty}^{\infty}a^{n(n+1)/2}b^{n(n-1)/2},$
###### §20.11(iii) Ramanujan’s Change of Base
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). … These results are called Ramanujan’s changes of base. …
##### 14: 9.6 Relations to Other Functions
9.6.1 $\zeta=\tfrac{2}{3}z^{3/2},$
9.6.10 $z=(\tfrac{3}{2}\zeta)^{2/3},$
9.6.11 $J_{\pm 1/3}\left(\zeta\right)=\tfrac{1}{2}\sqrt{3/z}\left(\sqrt{3}\mathrm{Ai}% \left(-z\right)\mp\mathrm{Bi}\left(-z\right)\right),$
9.6.13 $I_{\pm 1/3}\left(\zeta\right)=\tfrac{1}{2}\sqrt{3/z}\left(\mp\sqrt{3}\mathrm{% Ai}\left(z\right)+\mathrm{Bi}\left(z\right)\right),$
9.6.14 $I_{\pm 2/3}\left(\zeta\right)=\tfrac{1}{2}(\sqrt{3}/z)\left(\pm\sqrt{3}\mathrm% {Ai}'\left(z\right)+\mathrm{Bi}'\left(z\right)\right),$
##### 15: 27.12 Asymptotic Formulas: Primes
27.12.5 $\left|\pi\left(x\right)-\mathrm{li}\left(x\right)\right|=O\left(x\exp\left(-c(% \ln x)^{1/2}\right)\right),$ $x\to\infty$.
27.12.6 $\left|\pi\left(x\right)-\mathrm{li}\left(x\right)\right|=O\left(x\exp\left(-d(% \ln x)^{3/5}\,(\ln\ln x)^{-1/5}\right)\right).$
27.12.8 $\frac{\mathrm{li}\left(x\right)}{\phi\left(m\right)}+O\left(x\exp\left(-% \lambda(\alpha)(\ln x)^{1/2}\right)\right),$ $m\leq(\ln x)^{\alpha}$, $\alpha>0$,
##### 16: 1.17 Integral and Series Representations of the Dirac Delta
1.17.10 $\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-t^{2}/(4n)}e^{i(x-a)t}\mathrm{d}t=% \sqrt{\frac{n}{\pi}}e^{-n(x-a)^{2}}.$
1.17.11 $\delta_{n}\left(x-a\right)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-t^{2}/(4n)% }e^{i(x-a)t}\mathrm{d}t,$
1.17.23 $\delta\left(x-a\right)=e^{-(x+a)/2}\sum_{k=0}^{\infty}L_{k}\left(x\right)L_{k}% \left(a\right).$
1.17.24 $\delta\left(x-a\right)=\frac{e^{-(x^{2}+a^{2})/2}}{\sqrt{\pi}}\sum_{k=0}^{% \infty}\frac{H_{k}\left(x\right)H_{k}\left(a\right)}{2^{k}k!}.$
##### 17: 9.7 Asymptotic Expansions
9.7.1 $\zeta=\tfrac{2}{3}z^{\ifrac{3}{2}}.$
9.7.6 $\mathrm{Ai}'\left(z\right)\sim-\frac{z^{1/4}e^{-\zeta}}{2\sqrt{\pi}}\sum_{k=0}% ^{\infty}(-1)^{k}\frac{v_{k}}{\zeta^{k}},$ $|\operatorname{ph}z|\leq\pi-\delta$,
9.7.7 $\mathrm{Bi}\left(z\right)\sim\frac{e^{\zeta}}{\sqrt{\pi}z^{1/4}}\sum_{k=0}^{% \infty}\frac{u_{k}}{\zeta^{k}},$ $|\operatorname{ph}z|\leq\tfrac{1}{3}\pi-\delta$,
9.7.8 $\mathrm{Bi}'\left(z\right)\sim\frac{z^{1/4}e^{\zeta}}{\sqrt{\pi}}\sum_{k=0}^{% \infty}\frac{v_{k}}{\zeta^{k}},$ $|\operatorname{ph}z|\leq\tfrac{1}{3}\pi-\delta$.
9.7.13 $\mathrm{Bi}\left(ze^{\pm\pi i/3}\right)\mathrel{\sim}\sqrt{\frac{2}{\pi}}\frac% {e^{\pm\pi i/6}}{z^{1/4}}\*\left(\cos\left(\zeta-\tfrac{1}{4}\pi\mp\tfrac{1}{2% }\mathrm{i}\ln 2\right)\sum_{k=0}^{\infty}(-1)^{k}\frac{u_{2k}}{\zeta^{2k}}+% \sin\left(\zeta-\tfrac{1}{4}\pi\mp\tfrac{1}{2}\mathrm{i}\ln 2\right)\sum_{k=0}% ^{\infty}(-1)^{k}\frac{u_{2k+1}}{\zeta^{2k+1}}\right),$ $|\operatorname{ph}z|\leq\tfrac{2}{3}\pi-\delta$,
##### 18: 23.6 Relations to Other Functions
With $z=\ifrac{\pi u}{(2\omega_{1})}$, …
##### 19: 8.27 Approximations
• DiDonato (1978) gives a simple approximation for the function $F(p,x)=x^{-p}e^{x^{2}/2}\int_{x}^{\infty}e^{-t^{2}/2}t^{p}\mathrm{d}t$ (which is related to the incomplete gamma function by a change of variables) for real $p$ and large positive $x$. This takes the form $F(p,x)=4x/h(p,x)$, approximately, where $h(p,x)=3(x^{2}-p)+\sqrt{(x^{2}-p)^{2}+8(x^{2}+p)}$ and is shown to produce an absolute error $O\left(x^{-7}\right)$ as $x\to\infty$.

• ##### 20: 9.5 Integral Representations
9.5.7 $\mathrm{Ai}\left(z\right)=\frac{e^{-\zeta}}{\pi}\int_{0}^{\infty}\exp\left(-z^% {\ifrac{1}{2}}t^{2}\right)\cos\left(\tfrac{1}{3}t^{3}\right)\mathrm{d}t,$ $|\operatorname{ph}z|<\pi$.
9.5.8 $\mathrm{Ai}\left(z\right)=\frac{e^{-\zeta}\zeta^{\ifrac{-1}{6}}}{\sqrt{\pi}(48% )^{\ifrac{1}{6}}\Gamma\left(\frac{5}{6}\right)}\int_{0}^{\infty}e^{-t}t^{-% \ifrac{1}{6}}\left(2+\frac{t}{\zeta}\right)^{-\ifrac{1}{6}}\mathrm{d}t,$ $|\operatorname{ph}z|<\frac{2}{3}\pi$.