About the Project

coaxial%20circles

AdvancedHelp

(0.002 seconds)

1—10 of 471 matching pages

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. See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25). …
3: 20 Theta Functions
Chapter 20 Theta Functions
4: 25.12 Polylogarithms
25.12.4 Li 2 ( z ) + Li 2 ( 1 z ) = 1 6 π 2 1 2 ( ln ( z ) ) 2 , z [ 0 , ) .
25.12.6 Li 2 ( x ) + Li 2 ( 1 x ) = 1 6 π 2 ( ln x ) ln ( 1 x ) , 0 < x < 1 .
25.12.8 n = 1 cos ( n θ ) n 2 = π 2 6 π θ 2 + θ 2 4 .
See accompanying text
Figure 25.12.1: Dilogarithm function Li 2 ( x ) , 20 x < 1 . Magnify
See accompanying text
Figure 25.12.2: Absolute value of the dilogarithm function | Li 2 ( x + i y ) | , 20 x 20 , 20 y 20 . … Magnify 3D Help
5: 8 Incomplete Gamma and Related
Functions
6: 28 Mathieu Functions and Hill’s Equation
7: 8.26 Tables
  • Khamis (1965) tabulates P ( a , x ) for a = 0.05 ( .05 ) 10 ( .1 ) 20 ( .25 ) 70 , 0.0001 x 250 to 10D.

  • Abramowitz and Stegun (1964, pp. 245–248) tabulates E n ( x ) for n = 2 , 3 , 4 , 10 , 20 , x = 0 ( .01 ) 2 to 7D; also ( x + n ) e x E n ( x ) for n = 2 , 3 , 4 , 10 , 20 , x 1 = 0 ( .01 ) 0.1 ( .05 ) 0.5 to 6S.

  • Pagurova (1961) tabulates E n ( x ) for n = 0 ( 1 ) 20 , x = 0 ( .01 ) 2 ( .1 ) 10 to 4-9S; e x E n ( x ) for n = 2 ( 1 ) 10 , x = 10 ( .1 ) 20 to 7D; e x E p ( x ) for p = 0 ( .1 ) 1 , x = 0.01 ( .01 ) 7 ( .05 ) 12 ( .1 ) 20 to 7S or 7D.

  • Zhang and Jin (1996, Table 19.1) tabulates E n ( x ) for n = 1 , 2 , 3 , 5 , 10 , 15 , 20 , x = 0 ( .1 ) 1 , 1.5 , 2 , 3 , 5 , 10 , 20 , 30 , 50 , 100 to 7D or 8S.

  • 8: 23 Weierstrass Elliptic and Modular
    Functions
    9: 7.8 Inequalities
    7.8.2 1 x + x 2 + 2 < 𝖬 ( x ) 1 x + x 2 + ( 4 / π ) , x 0 ,
    7.8.3 π 2 π x + 2 𝖬 ( x ) < 1 x + 1 , x 0 ,
    7.8.5 x 2 2 x 2 + 1 x 2 ( 2 x 2 + 5 ) 4 x 4 + 12 x 2 + 3 x 𝖬 ( x ) < 2 x 4 + 9 x 2 + 4 4 x 4 + 20 x 2 + 15 < x 2 + 1 2 x 2 + 3 , x 0 .
    7.8.8 erf x < 1 e 4 x 2 / π , x > 0 .
    10: 22.3 Graphics
    See accompanying text
    Figure 22.3.13: sn ( x , k ) for k = 1 e n , n = 0 to 20, 5 π x 5 π . Magnify 3D Help
    See accompanying text
    Figure 22.3.14: cn ( x , k ) for k = 1 e n , n = 0 to 20, 5 π x 5 π . Magnify 3D Help
    See accompanying text
    Figure 22.3.15: dn ( x , k ) for k = 1 e n , n = 0 to 20, 5 π x 5 π . Magnify 3D Help
    See accompanying text
    Figure 22.3.28: Density plot of | sn ( 20 , k ) | as a function of complex k 2 , 10 ( k 2 ) 20 , 10 ( k 2 ) 10 . Grayscale, running from 0 (black) to 10 (white), with | sn ( 20 , k ) | > 10 truncated to 10. … Magnify