# 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: 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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##### 3: 31.15 Stieltjes Polynomials

##### 4: 10.51 Recurrence Relations and Derivatives

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##### 5: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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►If
${T}$
is a bounded operator then its spectrum is a closed bounded subset of
${\u2102}$
. If
${T}$
is self-adjoint (bounded or unbounded) then
${\sigma}{}{(}{T}{)}$
is a closed subset of
${\mathbb{R}}$
and the residual spectrum is empty.
Note that eigenfunctions for distinct (necessarily real) eigenvalues of
a self-adjoint operator are mutually orthogonal. If an eigenvalue is of multiplicity
greater than
${1}$
then an orthonormal basis of eigenfunctions can be given for the eigenspace.
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►
A

*boundary value*for the end point ${a}$ is a linear form ${\mathcal{B}}$ on ${\mathcal{D}}{}{(}{{\mathcal{L}}}^{{*}}{)}$ 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 reader is referred to Coddington and Levinson (1955), Friedman (1990, Ch. 3), Titchmarsh (1962a), and Everitt (2005b, pp. 45–74) and Everitt (2005a, pp. 272–331), for detailed methods and results. … ►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. This work is well overviewed by Coddington and Levinson (1955, Ch. 9), and then applied in detail by Titchmarsh (1946), Titchmarsh (1962a), Titchmarsh (1958), and Levitan and Sargsjan (1975) which also connects the Weyl theory to the relevant functional analysis. In parallel, similar, and more general formulations have grown out of functional analysis itself, as in the work of Stone (1990), Rudin (1973), Reed and Simon (1980), Reed and Simon (1975), Reed and Simon (1978), Reed and Simon (1979), Cycon et al. (2008), Dunford and Schwartz (1988, Ch. XIII), Hall (2013, pp. 127-223). Friedman (1990) provides a useful introduction to both approaches; as does the conference proceeding Amrein et al. (2005), overviewing the combination of Sturm–Liouville theory and Hilbert space theory. See, in particular, the overview Everitt (2005b, pp. 45–74), and the uniformly annotated listing of ${51}$ solved Sturm–Liouville problems in Everitt (2005a, pp. 272–331), each with their limit point, or circle, boundary behaviors categorized.##### 6: 10.6 Recurrence Relations and Derivatives

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►

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

##### 7: 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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##### 8: 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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##### 9: 10.29 Recurrence Relations and Derivatives

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##### 10: Bibliography R

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The effective resistance and inductance of a concentric main, and methods of computing the $\mathrm{ber}$ and $\mathrm{bei}$ and allied functions.
Philos. Mag. (6) 17, pp. 524–552.
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