# by parts

(0.002 seconds)

## 1—10 of 297 matching pages

##### 1: 26.11 Integer Partitions: Compositions
$c\left(n\right)$ denotes the number of compositions of $n$, and $c_{m}\left(n\right)$ is the number of compositions into exactly $m$ parts. $c\left(\in\!T,n\right)$ is the number of compositions of $n$ with no 1’s, where again $T=\{2,3,4,\ldots\}$. The integer 0 is considered to have one composition consisting of no parts: …
26.11.2 $c_{m}\left(0\right)=\delta_{0,m},$
26.11.3 $c_{m}\left(n\right)=\genfrac{(}{)}{0.0pt}{}{n-1}{m-1},$
##### 2: 27.4 Euler Products and Dirichlet Series
27.4.3 $\zeta\left(s\right)=\sum_{n=1}^{\infty}n^{-s}=\prod_{p}(1-p^{-s})^{-1},$ $\Re s>1$.
27.4.5 $\sum_{n=1}^{\infty}\mu\left(n\right)n^{-s}=\frac{1}{\zeta\left(s\right)},$ $\Re s>1$,
27.4.6 $\sum_{n=1}^{\infty}\phi\left(n\right)n^{-s}=\frac{\zeta\left(s-1\right)}{\zeta% \left(s\right)},$ $\Re s>2$,
27.4.11 $\sum_{n=1}^{\infty}\sigma_{\alpha}\left(n\right)n^{-s}=\zeta\left(s\right)% \zeta\left(s-\alpha\right),$ $\Re s>\max(1,1+\Re\alpha)$,
##### 3: 7.9 Continued Fractions
7.9.1 $\sqrt{\pi}e^{z^{2}}\operatorname{erfc}z=\cfrac{z}{z^{2}+\cfrac{\frac{1}{2}}{1+% \cfrac{1}{z^{2}+\cfrac{\frac{3}{2}}{1+\cfrac{2}{z^{2}+\cdots}}}}},$ $\Re z>0$,
7.9.2 $\sqrt{\pi}e^{z^{2}}\operatorname{erfc}z=\cfrac{2z}{2z^{2}+1-\cfrac{1\cdot 2}{2% z^{2}+5-\cfrac{3\cdot 4}{2z^{2}+9-\cdots}}},$ $\Re z>0$,
7.9.3 $w\left(z\right)=\frac{i}{\sqrt{\pi}}\cfrac{1}{z-\cfrac{\frac{1}{2}}{z-\cfrac{1% }{z-\cfrac{\frac{3}{2}}{z-\cfrac{2}{z-\cdots}}}}},$ $\Im z>0$.
##### 4: 8.14 Integrals
8.14.1 $\int_{0}^{\infty}e^{-ax}\frac{\gamma\left(b,x\right)}{\Gamma\left(b\right)}\,% \mathrm{d}x=\frac{(1+a)^{-b}}{a},$ $\Re a>0$, $\Re b>-1$,
8.14.2 $\int_{0}^{\infty}e^{-ax}\Gamma\left(b,x\right)\,\mathrm{d}x=\Gamma\left(b% \right)\frac{1-(1+a)^{-b}}{a},$ $\Re a>-1$, $\Re b>-1$.
8.14.3 $\int_{0}^{\infty}x^{a-1}\gamma\left(b,x\right)\,\mathrm{d}x=-\frac{\Gamma\left% (a+b\right)}{a},$ $\Re a<0$, $\Re\left(a+b\right)>0$,
8.14.4 $\int_{0}^{\infty}x^{a-1}\Gamma\left(b,x\right)\,\mathrm{d}x=\frac{\Gamma\left(% a+b\right)}{a},$ $\Re a>0$, $\Re\left(a+b\right)>0$,
8.14.6 $\int_{0}^{\infty}x^{a-1}e^{-sx}\Gamma\left(b,x\right)\,\mathrm{d}x=\frac{% \Gamma\left(a+b\right)}{a(1+s)^{a+b}}\*F\left(1,a+b;1+a;s/(1+s)\right),$ $\Re s>-1$, $\Re\left(a+b\right)>0$, $\Re a>0$.
Part 2. …
##### 6: 26.21 Tables
Abramowitz and Stegun (1964, Chapter 24) tabulates binomial coefficients $\genfrac{(}{)}{0.0pt}{}{m}{n}$ for $m$ up to 50 and $n$ up to 25; extends Table 26.4.1 to $n=10$; tabulates Stirling numbers of the first and second kinds, $s\left(n,k\right)$ and $S\left(n,k\right)$, for $n$ up to 25 and $k$ up to $n$; tabulates partitions $p\left(n\right)$ and partitions into distinct parts $p\left(\mathcal{D},n\right)$ for $n$ up to 500. Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts $\not\equiv\pm 2\pmod{5}$, partitions into parts $\not\equiv\pm 1\pmod{5}$, and unrestricted plane partitions up to 100. …
##### 7: 28.25 Asymptotic Expansions for Large $\Re z$
###### §28.25 Asymptotic Expansions for Large $\Re z$
28.25.4 $\Re z\to+\infty,$ $-\pi+\delta\leq\operatorname{ph}h+\Im z\leq 2\pi-\delta$,
28.25.5 $\Re z\to+\infty,$ $-2\pi+\delta\leq\operatorname{ph}h+\Im z\leq\pi-\delta$,
##### 8: 19.3 Graphics Figure 19.3.9: ℜ ⁡ ( K ⁡ ( k ) ) as a function of complex k 2 for − 2 ≤ ℜ ⁡ ( k 2 ) ≤ 2 , − 2 ≤ ℑ ⁡ ( k 2 ) ≤ 2 . The real part is symmetric under reflection in the real axis. … Magnify 3D Help Figure 19.3.10: ℑ ⁡ ( K ⁡ ( k ) ) as a function of complex k 2 for − 2 ≤ ℜ ⁡ ( k 2 ) ≤ 2 , − 2 ≤ ℑ ⁡ ( k 2 ) ≤ 2 . The imaginary part is 0 for k 2 < 1 , and is antisymmetric under reflection in the real axis. … Magnify 3D Help Figure 19.3.11: ℜ ⁡ ( E ⁡ ( k ) ) as a function of complex k 2 for − 2 ≤ ℜ ⁡ ( k 2 ) ≤ 2 , − 2 ≤ ℑ ⁡ ( k 2 ) ≤ 2 . The real part is symmetric under reflection in the real axis. … Magnify 3D Help Figure 19.3.12: ℑ ⁡ ( E ⁡ ( k ) ) as a function of complex k 2 for − 2 ≤ ℜ ⁡ ( k 2 ) ≤ 2 , − 2 ≤ ℑ ⁡ ( k 2 ) ≤ 2 . The imaginary part is 0 for k 2 ≤ 1 and is antisymmetric under reflection in the real axis. … Magnify 3D Help
##### 9: 13.10 Integrals
13.10.3 $\int_{0}^{\infty}e^{-zt}t^{b-1}{\mathbf{M}}\left(a,c,kt\right)\,\mathrm{d}t=% \Gamma\left(b\right)z^{-b}{{}_{2}{\mathbf{F}}_{1}}\left(a,b;c;\ifrac{k}{z}% \right),$ $\Re b>0$, $\Re z>\max\left(\Re k,0\right)$,
13.10.4 $\int_{0}^{\infty}e^{-zt}t^{b-1}{\mathbf{M}}\left(a,b,t\right)\,\mathrm{d}t=z^{% -b}\left(1-\frac{1}{z}\right)^{-a},$ $\Re b>0$, $\Re z>1$,
13.10.5 $\int_{0}^{\infty}e^{-t}t^{b-1}{\mathbf{M}}\left(a,c,t\right)\,\mathrm{d}t=% \frac{\Gamma\left(b\right)\Gamma\left(c-a-b\right)}{\Gamma\left(c-a\right)% \Gamma\left(c-b\right)},$ $\Re\left(c-a\right)>\Re b>0$,
13.10.7 $\int_{0}^{\infty}e^{-zt}t^{b-1}U\left(a,c,t\right)\,\mathrm{d}t=\Gamma\left(b% \right)\Gamma\left(b-c+1\right)\*z^{-b}{{}_{2}{\mathbf{F}}_{1}}\left(a,b;a+b-c% +1;1-\frac{1}{z}\right),$ $\Re b>\max\left(\Re c-1,0\right)$, $\Re z>0$.
13.10.10 $\int_{0}^{\infty}t^{\lambda-1}{\mathbf{M}}\left(a,b,-t\right)\,\mathrm{d}t=% \frac{\Gamma\left(\lambda\right)\Gamma\left(a-\lambda\right)}{\Gamma\left(a% \right)\Gamma\left(b-\lambda\right)},$ $0<\Re\lambda<\Re a$,
##### 10: 7.14 Integrals
7.14.2 $\int_{0}^{\infty}e^{-at}\operatorname{erf}\left(bt\right)\,\mathrm{d}t=\frac{1% }{a}e^{a^{2}/(4b^{2})}\operatorname{erfc}\left(\frac{a}{2b}\right),$ $\Re a>0$, $|\operatorname{ph}b|<\tfrac{1}{4}\pi$,
7.14.3 $\int_{0}^{\infty}e^{-at}\operatorname{erf}\sqrt{bt}\,\mathrm{d}t=\frac{1}{a}% \sqrt{\frac{b}{a+b}},$ $\Re a>0$, $\Re b>0$,
7.14.4 $\int_{0}^{\infty}e^{(a-b)t}\operatorname{erfc}\left(\sqrt{at}+\sqrt{\frac{c}{t% }}\right)\,\mathrm{d}t=\frac{e^{-2(\sqrt{ac}+\sqrt{bc})}}{\sqrt{b}(\sqrt{a}+% \sqrt{b})},$ $|\operatorname{ph}a|<\frac{1}{2}\pi$, $\Re b>0$, $\Re c\geq 0$.