# coaxial circles

(0.002 seconds)

## 1—10 of 400 matching pages

##### 1: 19.34 Mutual Inductance of Coaxial Circles
###### §19.34 Mutual Inductance of CoaxialCircles
The mutual inductance $M$ of two coaxial circles of radius $a$ and $b$ with centers at a distance $h$ apart is given in cgs units by
19.34.1 $\frac{{c}^{2}M}{2\pi}=ab\int_{0}^{2\pi}(h^{2}+a^{2}+b^{2}-2ab\cos\theta)^{-1/2% }\cos\theta\,\mathrm{d}\theta=2ab\int_{-1}^{1}\frac{t\,\mathrm{d}t}{\sqrt{(1+t% )(1-t)(a_{3}-2abt)}}=2abI(\mathbf{e}_{5}),$
is the square of the maximum (upper signs) or minimum (lower signs) distance between the circles. …
19.34.5 $\frac{3{c}^{2}}{8\pi ab}M=3R_{F}\left(0,r_{+}^{2},r_{-}^{2}\right)-2r_{-}^{2}R% _{D}\left(0,r_{+}^{2},r_{-}^{2}\right),$
##### 2: 10.73 Physical Applications
Consequently, Bessel functions $J_{n}\left(x\right)$, and modified Bessel functions $I_{n}\left(x\right)$, are central to the analysis of microwave and optical transmission in waveguides, including coaxial and fiber. …
##### 3: 4.22 Infinite Products and Partial Fractions
4.22.5 $\csc z=\frac{1}{z}+2z\sum_{n=1}^{\infty}\frac{(-1)^{n}}{z^{2}-n^{2}\pi^{2}}.$
##### 4: 4.36 Infinite Products and Partial Fractions
4.36.4 ${\operatorname{csch}}^{2}z=\sum_{n=-\infty}^{\infty}\frac{1}{(z-n\pi i)^{2}},$
##### 5: 3.12 Mathematical Constants
3.12.1 $\pi=3.14159\;26535\;89793\;23846\;\ldots$
##### 6: 6.15 Sums
6.15.1 $\sum_{n=1}^{\infty}\operatorname{Ci}\left(\pi n\right)=\tfrac{1}{2}(\ln 2-% \gamma),$
6.15.2 $\sum_{n=1}^{\infty}\frac{\operatorname{si}\left(\pi n\right)}{n}=\tfrac{1}{2}% \pi(\ln\pi-1),$
6.15.3 $\sum_{n=1}^{\infty}(-1)^{n}\operatorname{Ci}\left(2\pi n\right)=1-\ln 2-\tfrac% {1}{2}\gamma,$
6.15.4 $\sum_{n=1}^{\infty}(-1)^{n}\frac{\operatorname{si}\left(2\pi n\right)}{n}=\pi(% \tfrac{3}{2}\ln 2-1).$
##### 7: 18.33 Polynomials Orthogonal on the Unit Circle
###### §18.33(v) Biorthogonal Polynomials on the Unit Circle
See Al-Salam and Ismail (1994) for special biorthogonal rational functions on the unit circle.
##### 9: 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 $w\left(z\right)=\frac{i}{\sqrt{\pi}}\cfrac{1}{z-\cfrac{\frac{1}{2}}{z-\cfrac{1% }{z-\cfrac{\frac{3}{2}}{z-\cfrac{2}{z-\cdots}}}}},$ $\Im z>0$.
##### 10: 7.10 Derivatives
7.10.1 $\frac{{\mathrm{d}}^{n+1}\operatorname{erf}z}{{\mathrm{d}z}^{n+1}}=(-1)^{n}% \frac{2}{\sqrt{\pi}}H_{n}\left(z\right)e^{-z^{2}},$ $n=0,1,2,\dots$.