# Documentation

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# whittakerW

Whittaker W function

## Syntax

whittakerW(a,b,z)

## Description

whittakerW(a,b,z) returns the value of the Whittaker W function.

## Input Arguments

 a Symbolic number, variable, expression, function, or a vector or matrix of symbolic numbers, variables, expressions, or functions. If a is a vector or matrix, whittakerW returns the beta function for each element of a. b Symbolic number, variable, expression, function, or a vector or matrix of symbolic numbers, variables, expressions, or functions. If b is a vector or matrix, whittakerW returns the beta function for each element of b. z Symbolic number, variable, expression, function, or a vector or matrix of symbolic numbers, variables, expressions, or functions. If x is a vector or matrix, whittakerW returns the beta function for each element of z.

## Examples

Solve this second-order differential equation. The solutions are given in terms of the Whittaker functions.

syms a b w(z)
dsolve(diff(w, 2) + (-1/4 + a/z + (1/4 - b^2)/z^2)*w == 0)
ans =
C2*whittakerM(-a, -b, -z) + C3*whittakerW(-a, -b, -z)

Verify that the Whittaker W function is a valid solution of this differential equation:

syms a b z
isAlways(diff(whittakerW(a, b, z), z, 2) +...
(-1/4 + a/z + (1/4 - b^2)/z^2)*whittakerW(a, b, z) == 0)
ans =
logical
1

Verify that whittakerW(-a, -b, -z) also is a valid solution of this differential equation:

syms a b z
isAlways(diff(whittakerW(-a, -b, -z), z, 2) +...
(-1/4 + a/z + (1/4 - b^2)/z^2)*whittakerW(-a, -b, -z) == 0)
ans =
logical
1

Compute the Whittaker W function for these numbers. Because these numbers are not symbolic objects, you get floating-point results.

[whittakerW(1, 1, 1), whittakerW(-2, 1, 3/2 + 2*i),...
whittakerW(2, 2, 2), whittakerW(3, -0.3, 1/101)]
ans =
1.1953            -0.0156 - 0.0225i   4.8616            -0.1692

Compute the Whittaker W function for the numbers converted to symbolic objects. For most symbolic (exact) numbers, whittakerW returns unresolved symbolic calls.

[whittakerW(sym(1), 1, 1), whittakerW(-2, sym(1), 3/2 + 2*i),...
whittakerW(2, 2, sym(2)), whittakerW(sym(3), -0.3, 1/101)]
ans =
[ whittakerW(1, 1, 1), whittakerW(-2, 1, 3/2 + 2i),
whittakerW(2, 2, 2), whittakerW(3, -3/10, 1/101)]

For symbolic variables and expressions, whittakerW also returns unresolved symbolic calls:

syms a b x y
[whittakerW(a, b, x), whittakerW(1, x, x^2),...
whittakerW(2, x, y), whittakerW(3, x + y, x*y)]
ans =
[ whittakerW(a, b, x), whittakerW(1, x, x^2),
whittakerW(2, x, y), whittakerW(3, x + y, x*y)]

The Whittaker W function has special values for some parameters:

whittakerW(sym(-3/2), 1/2, 0)
ans =
4/(3*pi^(1/2))
syms a b x
whittakerW(0, b, x)
ans =
(x^(b + 1/2)*besselk(b, x/2))/(x^b*pi^(1/2))
whittakerW(a, -a + 1/2, x)
ans =
x^(1 - a)*x^(2*a - 1)*exp(-x/2)
whittakerW(a - 1/2, a, x)
ans =
(x^(a + 1/2)*exp(-x/2)*exp(x)*igamma(2*a, x))/x^(2*a)

Differentiate the expression involving the Whittaker W function:

syms a b z
diff(whittakerW(a,b,z), z)
ans =
- (a/z - 1/2)*whittakerW(a, b, z) -...
whittakerW(a + 1, b, z)/z

Compute the Whittaker W function for the elements of matrix A:

syms x
A = [-1, x^2; 0, x];
whittakerW(-1/2, 0, A)
ans =
[ -exp(-1/2)*(ei(1) + pi*1i)*1i,...
exp(x^2)*exp(-x^2/2)*expint(x^2)*(x^2)^(1/2)]
[  0,...
x^(1/2)*exp(-x/2)*exp(x)*expint(x)]

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### Whittaker W Function

The Whittaker functions Ma,b(z) and Wa,b(z) are linearly independent solutions of this differential equation:

$\frac{{d}^{2}w}{d{z}^{2}}+\left(-\frac{1}{4}+\frac{a}{z}+\frac{1/4-{b}^{2}}{{z}^{2}}\right)w=0$

The Whittaker W function is defined via the confluent hypergeometric functions:

${W}_{a,b}\left(z\right)={e}^{-z/2}{z}^{b+1/2}U\left(b-a+\frac{1}{2},1+2b,z\right)$

### Tips

• All non-scalar arguments must have the same size. If one or two input arguments are non-scalar, then whittakerW expands the scalars into vectors or matrices of the same size as the non-scalar arguments, with all elements equal to the corresponding scalar.

## References

Slater, L. J. "Cofluent Hypergeometric Functions." Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. (M. Abramowitz and I. A. Stegun, eds.). New York: Dover, 1972.

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