kummerU

Confluent hypergeometric Kummer U function

Syntax

kummerU(a,b,z)

Description

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:

$$U\left(a,b,z\right)=\frac{1}{\Gamma \left(a\right)}{\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-zt}{t}^{a-1}{\left(1+t\right)}^{b-a-1}dt}$$

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

a — Parameter of Kummer U function

number | vector | symbolic number | symbolic variable | symbolic expression | symbolic function | symbolic vector

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

b — Parameter of Kummer U function

number | vector | symbolic number | symbolic variable | symbolic expression | symbolic function | symbolic vector

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

z — Argument of Kummer U function

number | vector | symbolic number | symbolic variable | symbolic expression | symbolic function | symbolic 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.

More About

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:

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

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.

Version History

Introduced in R2014b