# mutual inductance of coaxial circles

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##### 1: 19.34 Mutual Inductance of Coaxial Circles

###### §19.34 Mutual Inductance of Coaxial Circles

►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
$$\frac{{c}^{2}M}{2\pi}=ab{\int}_{0}^{2\pi}{({h}^{2}+{a}^{2}+{b}^{2}-2ab\mathrm{cos}\theta )}^{-1/2}\mathrm{cos}\theta d\theta =2ab{\int}_{-1}^{1}\frac{tdt}{\sqrt{(1+t)(1-t)({a}_{3}-2abt)}}=2abI({\mathbf{e}}_{5}),$$

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►is the square of the maximum (upper signs) or minimum (lower signs) distance between the circles.
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►References for other inductance problems solvable in terms of elliptic integrals are given in Grover (1946, pp. 8 and 283).
##### 2: Bibliography G

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Mutual integrability, quadratic algebras, and dynamical symmetry.
Ann. Phys. 217 (1), pp. 1–20.
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Special functions of matrix argument. I. Algebraic induction, zonal polynomials, and hypergeometric functions.
Trans. Amer. Math. Soc. 301 (2), pp. 781–811.
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Inductance Calculations.
Van Nostrand, New York.
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##### 3: Tom M. Apostol

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►Tom Apostol and his wife Jane were inducted into the MAA’s Icosahedron Society in 2010.
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##### 4: 10.51 Recurrence Relations and Derivatives

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##### 5: 31.15 Stieltjes Polynomials

##### 6: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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►Note that eigenfunctions for distinct (necessarily real) eigenvalues of a self-adjoint operator are mutually orthogonal.
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►By

*Weyl’s alternative*${n}_{1}$ equals either 1 (the*limit point case*) or 2 (the*limit circle case*), and similarly for ${n}_{2}$. … A*boundary value*for the end point $a$ is a linear form $\mathcal{B}$ on $\mathcal{D}({\mathcal{L}}^{\ast})$ of the form … ►The above results, especially the discussions of deficiency indices and limit point and limit circle boundary conditions, lay the basis for further applications. … ►The materials developed here follow from the extensions of the Sturm–Liouville theory of second order ODEs as developed by Weyl, to include the*limit point and limit circle*singular cases. …##### 7: 10.6 Recurrence Relations and Derivatives

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10.6.10
$${p}_{\nu}{s}_{\nu}-{q}_{\nu}{r}_{\nu}=4/({\pi}^{2}ab).$$

##### 8: Frank W. J. Olver

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►Department of Commerce Gold Medal, the highest honorary award granted by the Department, and was inducted into the NIST Portrait Gallery of Distinguished Scientists, Engineers, and Administrators.
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##### 9: 23.2 Definitions and Periodic Properties

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23.2.14
$${\eta}_{3}{\omega}_{2}-{\eta}_{2}{\omega}_{3}={\eta}_{2}{\omega}_{1}-{\eta}_{1}{\omega}_{2}={\eta}_{1}{\omega}_{3}-{\eta}_{3}{\omega}_{1}=\frac{1}{2}\pi \mathrm{i}.$$

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##### 10: 10.29 Recurrence Relations and Derivatives

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