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In applied mathematics, the Kelvin functions berν(x) and beiν(x) are the real and imaginary parts, respectively, of

where x is real, and Jν(z), is the νth order Bessel function of the first kind. Similarly, the functions Kerν(x) and Keiν(x) are the real and imaginary parts, respectively, of

where Kν(z) is the νth order modified Bessel function of the second kind.

These functions are named after William Thomson, 1st Baron Kelvin.

While the Kelvin functions are defined as the real and imaginary parts of Bessel functions with x taken to be real, the functions can be analytically continued for complex arguments xe, 0 ≤ φ < 2π. With the exception of Bern(x) and Bein(x) for integral n, the Kelvin functions have a branch point at x = 0.

Below, Γ(z) is the gamma function and ψ(z) is the digamma function.

ber(x)Edit

 
ber(x) for x between 0 and 20.
 
  for x between 0 and 50.

For integers n, bern(x) has the series expansion

 

where Γ(z) is the gamma function. The special case ber0(x), commonly denoted as just ber(x), has the series expansion

 

and asymptotic series

 ,

where

 
 
 

bei(x)Edit

 
bei(x) for x between 0 and 20.
 
  for x between 0 and 50.

For integers n, bein(x) has the series expansion

 

The special case bei0(x), commonly denoted as just bei(x), has the series expansion

 

and asymptotic series

 

where α,  , and   are defined as for ber(x).

ker(x)Edit

 
ker(x) for x between 0 and 14.
 
  for x between 0 and 50.

For integers n, kern(x) has the (complicated) series expansion

 

The special case ker0(x), commonly denoted as just ker(x), has the series expansion

 

and the asymptotic series

 

where

 
 
 

kei(x)Edit

 
kei(x) for x between 0 and 14.
 
  for x between 0 and 50.

For integer n, kein(x) has the series expansion

 

The special case kei0(x), commonly denoted as just kei(x), has the series expansion

 

and the asymptotic series

 

where β, f2(x), and g2(x) are defined as for ker(x).

See alsoEdit

ReferencesEdit

  • Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 9". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 379. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
  • Olver, F. W. J.; Maximon, L. C. (2010), "Bessel functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248

External linksEdit

  • Weisstein, Eric W. "Kelvin Functions." From MathWorld—A Wolfram Web Resource. [1]
  • GPL-licensed C/C++ source code for calculating Kelvin functions at codecogs.com: [2]