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

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

```syms a b z
simplify(diff(whittakerW(-a, -b, -z), z, 2) +...
(-1/4 + a/z + (1/4 - b^2)/z^2)*whittakerW(-a, -b, -z)) == 0```
```ans =
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 + 2*i),
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))/(pi^(1/2)*x^b)```
`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)*(pi*i + ei(1))*i,
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)]```

expand all

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.