Whittaker Function Calculator

Enter parameters kappa, mu and positive z to compute the first kind M and second kind W Whittaker functions via confluent hypergeometric functions.

Input

Enter the parameters kappa, mu and a positive argument z to compute the first kind M and second kind W using confluent hypergeometric functions.

kappa (κ)

Any real number

mu (μ)

Avoid half integers

Positive real number

First kind Whittaker function M

0.9356558772

at kappa = 0.5, mu = 0.3, z = 1.5

Second kind W

0.6069081668

Kummer M(a;b;z)

1.4320692

Tricomi U(a;b;z)

0.92890401

Parameter a

0.3

Parameter b

1.6

Series terms

Plot of M against z

The horizontal axis z runs from 0 to 4. The orange dot marks the value at the entered z.

M and W for varying z

Values at several z including the one you entered. The highlighted row matches your input.

The first kind is M_kappa,mu(z) = e^(-z/2) z^(mu+1/2) M(mu-kappa+1/2; 2mu+1; z), where M is the Kummer confluent hypergeometric function.
The second kind is W_kappa,mu(z) = e^(-z/2) z^(mu+1/2) U(mu-kappa+1/2; 2mu+1; z), where U is the Tricomi function.
The Kummer function M(a;b;z) is evaluated by the series sum (a)_n/(b)_n z^n/n!. This series converges for every complex z, but for large z intermediate terms can grow and cause loss of precision.
Only positive z is accepted. The tool does not compute for z less than or equal to zero.
When b = 2mu+1 is close to an integer (mu a half integer), the linking formula for U becomes singular, so the calculation is skipped. Shift mu slightly away from integers and half integers.
If the series fails to converge or the value overflows, an error is shown.
The gamma function is evaluated with the Lanczos approximation, so extreme parameters may carry larger numerical error.

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