Kelvin Functions ber/bei/ker/kei Calculator

Enter x to evaluate the first-kind Kelvin functions ber(x) and bei(x) and the second-kind ker(x) and kei(x) by series, with a plot of ber and bei.

Input

Enter a positive number x to compute the first-kind Kelvin functions ber(x) and bei(x) and the second-kind functions ker(x) and kei(x) via series expansion.

Argument x (positive)

ker and kei are undefined for x not greater than 0, so enter a positive value.

Result

ber(1)

0.9843817812

bei(1)

0.24956604

ker(1)

0.2867062087

kei(1)

-0.4949946365

Graph of ber(x) and bei(x)

ber(x)

bei(x)

How it works

ber(x) and bei(x) are computed from power series in (x/2): ber(x)=Σ(-1)^k (x/2)^(4k)/[(2k)!]^2 and bei(x)=Σ(-1)^k (x/2)^(4k+2)/[(2k+1)!]^2.
ker(x) and kei(x) add a logarithmic term with -ln(x/2) and a (π/4) term to an extra series whose coefficients are harmonic numbers. Because ker and kei are undefined for x not greater than 0, only positive x is accepted.
These functions arise from the first-kind Bessel function Jₙ evaluated at the complex argument x·e^(3πi/4): ber and bei give ber(x)+i·bei(x), while ker and kei correspond to the second kind.
The series is truncated once the relative size of the terms becomes small enough. For large x the values grow and oscillate quickly, so mind the rounding of displayed digits.

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