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# whittakerM

Whittaker M function

## Syntax

```whittakerM(a,b,z) ```

## Description

`whittakerM(a,b,z)` returns the value of the Whittaker M 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, `whittakerM` 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, `whittakerM` 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, `whittakerM` 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 M function is a valid solution of this differential equation:

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

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

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

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

```[whittakerM(1, 1, 1), whittakerM(-2, 1, 3/2 + 2*i),... whittakerM(2, 2, 2), whittakerM(3, -0.3, 1/101)]```
```ans = 0.7303 -9.2744 + 5.4705i 2.6328 0.3681```

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

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

For symbolic variables and expressions, `whittakerM` also returns unresolved symbolic calls:

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

The Whittaker M function has special values for some parameters:

`whittakerM(sym(-3/2), 1, 1)`
```ans = exp(1/2)```
```syms a b x whittakerM(0, b, x)```
```ans = 4^b*x^(1/2)*gamma(b + 1)*besseli(b, x/2)```
`whittakerM(a + 1/2, a, x)`
```ans = x^(a + 1/2)*exp(-x/2)```
`whittakerM(a, a - 5/2, x)`
```ans = (2*x^(a - 2)*exp(-x/2)*(2*a^2 - 7*a + x^2/2 -... x*(2*a - 3) + 6))/pochhammer(2*a - 4, 2)```

Differentiate the expression involving the Whittaker M function:

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

Compute the Whittaker M function for the elements of matrix `A`:

```syms x A = [-1, x^2; 0, x]; whittakerM(-1/2, 0, A)```
```ans = [ exp(-1/2)*1i, exp(x^2/2)*(x^2)^(1/2)] [ 0, x^(1/2)*exp(x/2)]```

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### Whittaker M 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 M function is defined via the confluent hypergeometric functions:

`${M}_{a,b}\left(z\right)={e}^{-z/2}\text{\hspace{0.17em}}{z}^{b+1/2}\text{\hspace{0.17em}}M\left(b-a+\frac{1}{2},\text{\hspace{0.17em}}1+2b,\text{\hspace{0.17em}}z\right)$`

## Tips

• All non-scalar arguments must have the same size. If one or two input arguments are non-scalar, then `whittakerM` 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.