kummerU

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))