Confluent hypergeometric Kummer U function
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
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 - 0.3204327531iThe 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^3If 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
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))
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.
[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.