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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 c 2 M 2 π = a b 0 2 π ( h 2 + a 2 + b 2 2 a b cos θ ) 1 / 2 cos θ d θ = 2 a b 1 1 t d t ( 1 + t ) ( 1 t ) ( a 3 2 a b t ) = 2 a b I ( 𝐞 5 ) ,
is the square of the maximum (upper signs) or minimum (lower signs) distance between the circles. …
19.34.5 3 c 2 8 π a b M = 3 R F ( 0 , r + 2 , r 2 ) 2 r 2 R D ( 0 , r + 2 , r 2 ) ,
2: 10.73 Physical Applications
Consequently, Bessel functions J n ( x ) , and modified Bessel functions I n ( x ) , are central to the analysis of microwave and optical transmission in waveguides, including coaxial and fiber. …
3: 4.22 Infinite Products and Partial Fractions
4.22.1 sin z = z n = 1 ( 1 z 2 n 2 π 2 ) ,
4.22.2 cos z = n = 1 ( 1 4 z 2 ( 2 n 1 ) 2 π 2 ) .
4.22.3 cot z = 1 z + 2 z n = 1 1 z 2 n 2 π 2 ,
4.22.4 csc 2 z = n = 1 ( z n π ) 2 ,
4.22.5 csc z = 1 z + 2 z n = 1 ( 1 ) n z 2 n 2 π 2 .
4: 4.36 Infinite Products and Partial Fractions
4.36.1 sinh z = z n = 1 ( 1 + z 2 n 2 π 2 ) ,
4.36.2 cosh z = n = 1 ( 1 + 4 z 2 ( 2 n 1 ) 2 π 2 ) .
4.36.3 coth z = 1 z + 2 z n = 1 1 z 2 + n 2 π 2 ,
4.36.4 csch 2 z = n = 1 ( z n π i ) 2 ,
4.36.5 csch z = 1 z + 2 z n = 1 ( 1 ) n z 2 + n 2 π 2 .
5: 3.12 Mathematical Constants
3.12.1 π = 3.14159 26535 89793 23846
3.12.2 π = 4 0 1 d t 1 + t 2 .
6: 6.15 Sums
6.15.2 n = 1 si ( π n ) n = 1 2 π ( ln π 1 ) ,
6.15.3 n = 1 ( 1 ) n Ci ( 2 π n ) = 1 ln 2 1 2 γ ,
6.15.4 n = 1 ( 1 ) n si ( 2 π n ) n = π ( 3 2 ln 2 1 ) .
7: 18.33 Polynomials Orthogonal on the Unit Circle
§18.33 Polynomials Orthogonal on the Unit Circle
§18.33(i) Definition
§18.33(ii) Recurrence Relations
§18.33(v) Biorthogonal Polynomials on the Unit Circle
See Al-Salam and Ismail (1994) for special biorthogonal rational functions on the unit circle.
8: 28.3 Graphics
9: 7.9 Continued Fractions
7.9.1 π e z 2 erfc z = z z 2 + 1 2 1 + 1 z 2 + 3 2 1 + 2 z 2 + , z > 0 ,
7.9.2 π e z 2 erfc z = 2 z 2 z 2 + 1 1 2 2 z 2 + 5 3 4 2 z 2 + 9 , z > 0 ,
7.9.3 w ( z ) = i π 1 z 1 2 z 1 z 3 2 z 2 z , z > 0 .
10: 7.10 Derivatives
7.10.1 d n + 1 erf z d z n + 1 = ( 1 ) n 2 π H n ( z ) e z 2 , n = 0 , 1 , 2 , .
7.10.2 w ( z ) = 2 z w ( z ) + ( 2 i / π ) ,