# kummerU

Confluent hypergeometric Kummer U function

## Syntax

``kummerU(a,b,z)``

## Description

example

````kummerU(a,b,z)` computes the value of confluent hypergeometric function, `U(a,b,z)`. If the real parts of `z` and `a` are positive values, then the integral representations of the Kummer U function is as follows:$U\left(a,b,z\right)=\frac{1}{\Gamma \left(a\right)}\underset{0}{\overset{\infty }{\int }}{e}^{-zt}{t}^{a-1}{\left(1+t\right)}^{b-a-1}dt$```

## Examples

### Equation Returning the Kummer U Function as Its Solution

`dsolve` can return solutions of second-order ordinary differential equations in terms of the Kummer U function.

Solve this equation. The solver returns the results in terms of the Kummer U function and another hypergeometric function.

```syms t z y(z) dsolve(z^3*diff(y,2) + (z^2 + t)*diff(y) + z*y)```
```ans = (C4*hypergeom(1i/2, 1 + 1i, t/(2*z^2)))/z^1i +... (C3*kummerU(1i/2, 1 + 1i, t/(2*z^2)))/z^1i```

### Kummer U Function for Numeric and Symbolic Arguments

Depending on its arguments, `kummerU` can return floating-point or exact symbolic results.

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

```A = [kummerU(-1/3, 2.5, 2) kummerU(1/3, 2, pi) kummerU(1/2, 1/3, 3*i)]```
```A = 0.8234 + 0.0000i 0.7284 + 0.0000i 0.4434 - 0.3204i```

Compute the Kummer U function for the numbers converted to symbolic objects. For most symbolic (exact) numbers, `kummerU` returns unresolved symbolic calls.

```symA = [kummerU(-1/3, 2.5, sym(2)) kummerU(1/3, 2, sym(pi)) kummerU(1/2, sym(1/3), 3*i)]```
```symA = kummerU(-1/3, 5/2, 2) kummerU(1/3, 2, pi) kummerU(1/2, 1/3, 3i)```

Use `vpa` to approximate symbolic results with the required number of digits.

`vpa(symA,10)`
```ans = 0.8233667846 0.7284037305 0.4434362538 - 0.3204327531i```

### Some Special Values of Kummer U

The Kummer U function has special values for some parameters.

If `a` is a negative integer, the Kummer U function reduces to a polynomial.

```syms a b z [kummerU(-1, b, z) kummerU(-2, b, z) kummerU(-3, b, z)]```
```ans = z - b b - 2*z*(b + 1) + b^2 + z^2 6*z*(b^2/2 + (3*b)/2 + 1) - 2*b - 6*z^2*(b/2 + 1) - 3*b^2 - b^3 + z^3```

If `b = 2*a`, the Kummer U function reduces to an expression involving the modified Bessel function of the second kind.

`kummerU(a, 2*a, z)`
```ans = (z^(1/2 - a)*exp(z/2)*besselk(a - 1/2, z/2))/pi^(1/2)```

If `a = 1` or `a = b`, the Kummer U function reduces to an expression involving the incomplete gamma function.

`kummerU(1, b, z)`
```ans = z^(1 - b)*exp(z)*igamma(b - 1, z)```
`kummerU(a, a, z)`
```ans = exp(z)*igamma(1 - a, z)```

If `a = 0`, the Kummer U function is `1`.

`kummerU(0, a, z)`
```ans = 1```

### Handle Expressions Containing the Kummer U Function

Many functions, such as `diff`, `int`, and `limit`, can handle expressions containing `kummerU`.

Find the first derivative of the Kummer U function with respect to `z`.

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

Find the indefinite integral of the Kummer U function with respect to `z`.

`int(kummerU(a, b, z), z)`
```ans = ((b - 2)/(a - 1) - 1)*kummerU(a, b, z) +... (kummerU(a + 1, b, z)*(a - a*b + a^2))/(a - 1) -... (z*kummerU(a, b, z))/(a - 1) ```

Find the limit of this Kummer U function.

`limit(kummerU(1/2, -1, z), z, 0)`
```ans = 4/(3*pi^(1/2))```

## Input Arguments

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Parameter of Kummer U function, specified as a number, variable, symbolic expression, symbolic function, or vector.

Parameter of Kummer U function, specified as a number, variable, symbolic expression, symbolic function, or vector.

Argument of Kummer U function, specified as a number, variable, symbolic expression, symbolic function, or vector. If `z` is a vector, `kummerU(a,b,z)` is evaluated element-wise.

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### Confluent Hypergeometric Function (Kummer U Function)

The confluent hypergeometric function (Kummer U function) is one of the solutions of the differential equation

`$z\frac{{\partial }^{2}}{\partial {z}^{2}}y+\left(b-z\right)\frac{\partial }{\partial z}y-ay=0$`

The other solution is the hypergeometric function 1F1(a,b,z).

The Whittaker W function can be expressed in terms of the Kummer U function:

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

## Tips

• `kummerU` returns floating-point results for numeric arguments that are not symbolic objects.

• `kummerU` acts element-wise on nonscalar inputs.

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

## References

[1] Slater, L. J. “Confluent Hypergeometric Functions.” Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. (M. Abramowitz and I. A. Stegun, eds.). New York: Dover, 1972.