# change of modulus

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## 21—30 of 88 matching pages

##### 21: 23.6 Relations to Other Functions
With $z=\ifrac{\pi u}{(2\omega_{1})}$, …
##### 22: 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 $x\to\infty$.

• ##### 23: 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 $\mathrm{Ai}\left(z\right)=\frac{e^{-\zeta}\zeta^{\ifrac{-1}{6}}}{\sqrt{\pi}(48% )^{\ifrac{1}{6}}\Gamma\left(\frac{5}{6}\right)}\int_{0}^{\infty}e^{-t}t^{-% \ifrac{1}{6}}\left(2+\frac{t}{\zeta}\right)^{-\ifrac{1}{6}}\mathrm{d}t,$ $|\operatorname{ph}z|<\frac{2}{3}\pi$.
##### 24: 27.2 Functions
Gauss and Legendre conjectured that $\pi\left(x\right)$ is asymptotic to $x/\ln x$ as $x\to\infty$:
27.2.3 $\pi\left(x\right)\sim\frac{x}{\ln x}.$
27.2.4 $p_{n}\sim n\ln n.$
27.2.14 $\Lambda\left(n\right)=\ln p,$ $n=p^{a}$,
##### 25: 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 $k\sin\phi$, $-i\tan\phi$, $-ik^{\prime}\tan\phi$, $(k^{\prime}\sin\phi)/\sqrt{1-k^{2}{\sin^{2}}\phi}$, $-ik\sin\phi/\sqrt{1-k^{2}{\sin^{2}}\phi}$. …
##### 26: Errata
• Other Changes

• In Paragraph Steed’s Algorithm in §3.10(iii), a sentence was added to inform the reader of alternatives to Steed’s algorithm, namely the Lentz algorithm (see e.g., Lentz (1976)) and the modified Lentz algorithm (see e.g., Thompson and Barnett (1986)).

• In Subsection 19.11(i), a sentence and unnumbered equation

$R_{C}\left(\gamma-\delta,\gamma\right)=\frac{-1}{\sqrt{\delta}}\operatorname{% arctan}\left(\frac{\sqrt{\delta}\sin\theta\sin\phi\sin\psi}{\alpha^{2}-1-% \alpha^{2}\cos\theta\cos\phi\cos\psi}\right),$

were added which indicate that care must be taken with the multivalued functions in (19.11.5). See (Cayley, 1961, pp. 103-106). This was suggested by Albert Groenenboom.

• In Subsection 33.14(iv), there were several modifications. Just below (33.14.9), the constraint described in the text “$\ell<(-\epsilon)^{-1/2}$ when $\epsilon<0$,” was removed. In Equation (33.14.13), the constraint $\epsilon_{1},\epsilon_{2}>0$ was added. In the line immediately below (33.14.13), it was clarified that $s\left(\epsilon,\ell;r\right)$ is $\exp\left(-r/n\right)$ times a polynomial in $r/n$, instead of simply a polynomial in $r$. In Equation (33.14.14), a second equality was added which relates $\phi_{n,\ell}(r)$ to Laguerre polynomials. A sentence was added immediately below (33.14.15) indicating that the functions $\phi_{n,\ell}$, $n=\ell,\ell+1,\ldots$, do not form a complete orthonormal system.

• Other Changes

• The factor on the right-hand side of Equation (10.9.26) containing $\cos(\mu-\nu)\theta$ has been been replaced with $\cos\left((\mu-\nu)\theta\right)$ to clarify the meaning.

• In Paragraph Confluent Hypergeometric Functions in §10.16, several Whittaker confluent hypergeometric functions were incorrectly linked to the definitions of the Kummer confluent hypergeometric and parabolic cylinder functions. However, to the eye, the functions appeared correct. The links were corrected.

• In Equation (15.6.9), it was clarified that $\lambda\in\mathbb{C}$.

• Originally Equation (19.16.9) had the constraint $a,a^{\prime}>0$. This constraint was replaced with $b_{1}+\cdots+b_{n}>a>0$, $b_{j}\in\mathbb{R}$. It therefore follows from Equation (19.16.10) that $a^{\prime}>0$. The last sentence of Subsection 19.16(ii) was elaborated to mention that generalizations may also be found in Carlson (1977b). These were suggested by Bastien Roucariès.

• In Section 19.25(vi), the Weierstrass lattice roots $e_{j},$ were labeled inadvertently as the base of the natural logarithm. In order to resolve this inconsistency, the lattice roots $e_{j}$, and lattice invariants $g_{2}$, $g_{3}$, now link to their respective definitions (see §§23.2(i), 23.3(i)). This was reported by Felix Ospald.

• In Equation (19.25.37), the Weierstrass zeta function was incorrectly linked to the definition of the Riemann zeta function. However, to the eye, the function appeared correct. The link was corrected.

• In Equation (27.12.5), the term originally written as $\sqrt{\ln x}$ was rewritten as $(\ln x)^{1/2}$ to be consistent with other equations in the same subsection.

• Other Changes

• In §5.2(iii), three new identities for Pochhammer’s symbol (5.2.6)–(5.2.8) have been added at the end of this subsection. This was suggested by Tom Koornwinder.

• Originally named as a complementary error function, (7.2.3) has been renamed as the Faddeeva (or Faddeyeva) function $w\left(z\right)$.

• In §7.8, an inequality (7.8.8) was added at the end of this section. This is Pólya (1949, (1.5)) and was suggested by Roberto Iacono.

• Originally the function $\chi$, used in (9.7.3) and (9.7.4), was presented with argument given by a positive integer $n$. It has now been clarified to be valid for argument given by a positive real number $x$.

• Bounds have been sharpened in §9.7(iii). The second paragraph now reads, “The $n$th error term is bounded in magnitude by the first neglected term multiplied by $\chi(n+\sigma)+1$ where $\sigma=\frac{1}{6}$ for (9.7.7) and $\sigma=0$ for (9.7.8), provided that $n\geq 0$ in the first case and $n\geq 1$ in the second case.” Previously it read, “In (9.7.7) and (9.7.8) the $n$th error term is bounded in magnitude by the first neglected term multiplied by $2\chi(n)\exp\left(\sigma\pi/(72\zeta)\right)$ where $\sigma=5$ for (9.7.7) and $\sigma=7$ for (9.7.8), provided that $n\geq 1$ in both cases.” In Equation (9.7.16)

9.7.16
$\mathrm{Bi}\left(x\right)\leq\frac{{\mathrm{e}^{\xi}}}{\sqrt{\pi}x^{1/4}}\left% (1+\left(\chi(\tfrac{7}{6})+1\right)\frac{5}{72\xi}\right),$
$\mathrm{Bi}'\left(x\right)\leq\frac{x^{1/4}{\mathrm{e}^{\xi}}}{\sqrt{\pi}}% \left(1+\left(\frac{\pi}{2}+1\right)\frac{7}{72\xi}\right),$

the bounds on the right-hand sides have been sharpened. The factors $\left(\chi(\tfrac{7}{6})+1\right)\frac{5}{72\xi}$, $\left(\frac{\pi}{2}+1\right)\frac{7}{72\xi}$, were originally given by $\frac{5\pi}{72\xi}\exp\left(\frac{5\pi}{72\xi}\right)$, $\frac{7\pi}{72\xi}\exp\left(\frac{7\pi}{72\xi}\right)$, respectively.

• Bounds have been sharpened in §9.7(iv). The first paragraph now reads, “The $n$th error term in (9.7.5) and (9.7.6) is bounded in magnitude by the first neglected term multiplied by

9.7.17
$\begin{cases}1,&|\operatorname{ph}z|\leq\tfrac{1}{3}\pi,\\ \min\left(|\!\csc\left(\operatorname{ph}\zeta\right)|,\chi(n\!+\!\sigma)\!+\!1% \right),&\tfrac{1}{3}\pi\leq|\operatorname{ph}z|\leq\tfrac{2}{3}\pi,\\ \frac{\sqrt{2\pi(n\!+\!\sigma)}}{|\!\cos\left(\operatorname{ph}\zeta\right)|^{% n\!+\!\sigma}}+\chi(n\!+\!\sigma)\!+\!1,&\tfrac{2}{3}\pi\leq|\operatorname{ph}% z|<\pi,\end{cases}$

provided that $n\geq 0$, $\sigma=\tfrac{1}{6}$ for (9.7.5) and $n\geq 1$, $\sigma=0$ for (9.7.6).” Previously it read, “When $n\geq 1$ the $n$th error term in (9.7.5) and (9.7.6) is bounded in magnitude by the first neglected term multiplied by

9.7.17
$\begin{cases}2\exp\left(\dfrac{\sigma}{36|\zeta|}\right)&|\operatorname{ph}z|% \leq\tfrac{1}{3}\pi,\\ 2\chi(n)\exp\left(\dfrac{\sigma\pi}{72|\zeta|}\right)&\tfrac{1}{3}\pi\leq|% \operatorname{ph}z|\leq\tfrac{2}{3}\pi,\\ \dfrac{4\chi(n)}{|\cos\left(\operatorname{ph}\zeta\right)|^{n}}\exp\left(% \dfrac{\sigma\pi}{36|\Re\zeta|}\right)&\tfrac{2}{3}\pi\leq|\operatorname{ph}z|% <\pi.\end{cases}$

Here $\sigma=5$ for (9.7.5) and $\sigma=7$ for (9.7.6).”

• In §10.8, a sentence was added just below (10.8.3) indicating that it is a rewriting of (16.12.1). This was suggested by Tom Koornwinder.

• Equations (10.15.1), (10.38.1), have been generalized to include the additional cases of $\ifrac{\partial J_{-\nu}\left(z\right)}{\partial\nu}$, $\ifrac{\partial I_{-\nu}\left(z\right)}{\partial\nu}$, respectively.

• The Kronecker delta symbols in Equations (10.22.37), (10.22.38), (14.17.6)–(14.17.9), have been moved furthest to the right, as is common convention for orthogonality relations.

• The titles of §§14.5(ii), 14.5(vi), have been changed to $\mu=0$ , $\nu=0,1$ , and Addendum to §14.5(ii)$\mu=0$, $\nu=2$ , respectively, in order to be more descriptive of their contents.

• The second and the fourth lines of (19.7.2) containing $k^{\prime}/ik$ have both been replaced with $-ik^{\prime}/k$ to clarify the meaning.

• Originally Equation (25.2.4) had the constraint $\Re s>0$. This constraint was removed because, as stated after (25.2.1), $\zeta\left(s\right)$ is meromorphic with a simple pole at $s=1$, and therefore $\zeta\left(s\right)-(s-1)^{-1}$ is an entire function. This was suggested by John Harper.

• The title of §32.16 was changed from Physical to Physical Applications.

• Bibliographic citations and clarifications have been added, removed, or modified in §§5.6(i), 5.10, 7.8, 7.25(iii), and 32.16.

• Other Changes

• To be consistent with the notation used in (8.12.16), Equation (8.12.5) was changed to read

8.12.5 $\frac{{\mathrm{e}^{\pm\pi\mathrm{i}a}}}{2\mathrm{i}\sin(\pi a)}Q\left(-a,z{% \mathrm{e}^{\pm\pi\mathrm{i}}}\right)=\pm\tfrac{1}{2}\operatorname{erfc}\left(% \pm\mathrm{i}\eta\sqrt{a/2}\right)-\mathrm{i}T(a,\eta)$
• Following a suggestion from James McTavish on 2017-04-06, the recurrence relation $u_{k}=\frac{(6k-5)(6k-3)(6k-1)}{(2k-1)216k}u_{k-1}$ was added to Equation (9.7.2).

• In §15.2(ii), the unnumbered equation

$\lim_{c\to-n}\frac{F\left(a,b;c;z\right)}{\Gamma\left(c\right)}=\mathbf{F}% \left(a,b;-n;z\right)=\frac{{\left(a\right)_{n+1}}{\left(b\right)_{n+1}}}{(n+1% )!}z^{n+1}F\left(a+n+1,b+n+1;n+2;z\right),$ $n=0,1,2,\dots$

was added in the second paragraph. An equation number will be assigned in an expanded numbering scheme that is under current development. Additionally, the discussion following (15.2.6) was expanded.

• In §15.4(i), due to a report by Louis Klauder on 2017-01-01, and in §15.4(iii), sentences were added specifying that some equations in these subsections require special care under certain circumstances. Also, (15.4.6) was expanded by adding the formula $F\left(a,b;a;z\right)=(1-z)^{-b}$.

• A bibliographic citation was added in §11.13(i).

• Other Changes

• ##### 27: 2.5 Mellin Transform Methods
2.5.9 $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(1-z\right)=\frac{\pi}{\sin\left(% \pi z\right)},$ $0<\Re z<1$,
2.5.10 $\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z\right)=\frac{2^{z-1}\Gamma\left(% \nu+\frac{1}{2}z\right)}{{\Gamma^{2}}\left(1-\frac{1}{2}z\right)\Gamma\left(1+% \nu-\frac{1}{2}z\right)\Gamma\left(z\right)}\frac{\pi}{\sin\left(\pi z\right)},$ $-2\nu<\Re z<1$.
2.5.11 $\Residue_{z=n}\left[x^{-z}\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(1-z% \right)\mathscr{M}\mskip-3.0mu h\mskip 3.0mu \left(z\right)\right]=(a_{n}\ln x% +b_{n})x^{-n},$
##### 28: 1.14 Integral Transforms
1.14.17 $\mathscr{L}\left(f\right)\left(s\right)=\mathscr{L}\mskip-3.0mu f\mskip 3.0mu % \left(s\right)=\int^{\infty}_{0}e^{-st}f(t)\mathrm{d}t.$
1.14.19 $\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(s\right)\to 0,$ $\Re s\to\infty$.
If also $\lim_{t\to 0+}f(t)/t$ exists, then … where $A_{p}=\tan\left(\tfrac{1}{2}\pi/p\right)$ when $1, or $\cot\left(\tfrac{1}{2}\pi/p\right)$ when $p\geq 2$. …
1.14.46 $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\mathcal{H}\mskip-3.0mu f\mskip 3.% 0mu \left(u\right)e^{iux}\mathrm{d}u=-\mathrm{i}(\operatorname{sign}x)\mathscr% {F}\mskip-3.0mu f\mskip 3.0mu \left(x\right),$
##### 29: 15.12 Asymptotic Approximations
15.12.5 $\mathbf{F}\left({a+\lambda,b-\lambda\atop c};\tfrac{1}{2}-\tfrac{1}{2}z\right)% =2^{(a+b-1)/2}\frac{(z+1)^{(c-a-b-1)/2}}{(z-1)^{c/2}}\sqrt{\zeta\sinh\zeta}% \left(\lambda+\tfrac{1}{2}a-\tfrac{1}{2}b\right)^{1-c}\left(I_{c-1}\left((% \lambda+\tfrac{1}{2}a-\tfrac{1}{2}b)\zeta\right)(1+O(\lambda^{-2}))+\frac{I_{c% -2}\left((\lambda+\tfrac{1}{2}a-\tfrac{1}{2}b)\zeta\right)}{2\lambda+a-b}\left% (\left(c-\tfrac{1}{2}\right)\left(c-\tfrac{3}{2}\right)\left(\frac{1}{\zeta}-% \coth\zeta\right)+\tfrac{1}{2}(2c-a-b-1)(a+b-1)\tanh\left(\tfrac{1}{2}\zeta% \right)+O(\lambda^{-2})\right)\right),$
15.12.9 $(z+1)^{3\lambda/2}(2\lambda)^{c-1}\mathbf{F}\left({a+\lambda,b+2\lambda\atop c% };-z\right)={\lambda^{-1/3}\left(e^{\pi\mathrm{i}(a-c+\lambda+(1/3))}\mathrm{% Ai}\left(e^{-\ifrac{2\pi\mathrm{i}}{3}}\lambda^{\ifrac{2}{3}}\beta^{2}\right)+% e^{\pi\mathrm{i}(c-a-\lambda-(1/3))}\mathrm{Ai}\left(e^{\ifrac{2\pi\mathrm{i}}% {3}}\lambda^{\ifrac{2}{3}}\beta^{2}\right)\right)\left(a_{0}(\zeta)+O(\lambda^% {-1})\right)}+\lambda^{-2/3}\left(e^{\pi\mathrm{i}(a-c+\lambda+(2/3))}\mathrm{% Ai}'\left(e^{-\ifrac{2\pi\mathrm{i}}{3}}\lambda^{\ifrac{2}{3}}\beta^{2}\right)% +e^{\pi\mathrm{i}(c-a-\lambda-(2/3))}\mathrm{Ai}'\left(e^{\ifrac{2\pi\mathrm{i% }}{3}}\lambda^{\ifrac{2}{3}}\beta^{2}\right)\right)\left(a_{1}(\zeta)+O(% \lambda^{-1})\right),$
15.12.11 $\beta=\left(-\frac{3}{2}\zeta+\frac{9}{4}\ln\left(\frac{2+e^{\zeta}}{2+e^{-% \zeta}}\right)\right)^{1/3},$
15.12.13 $G_{0}(\pm\beta)=\left(2+e^{\pm\zeta}\right)^{c-b-(\ifrac{1}{2})}\left(1+e^{\pm% \zeta}\right)^{a-c+(\ifrac{1}{2})}\left(z-1-e^{\pm\zeta}\right)^{-a+(\ifrac{1}% {2})}\sqrt{\frac{\beta}{e^{\zeta}-e^{-\zeta}}}.$
##### 30: 19.2 Definitions
The cases with $\phi=\pi/2$ are the complete integrals: …
$k_{c}=k^{\prime},$
$p=1-\alpha^{2},$
$x=\tan\phi,$
If $1, then $k_{c}$ is pure imaginary. …