Hypergeometric function

The hypergeometric function is in mathematics a function that generalizes the geometric series. You will be counted to the class of special functions and is a solution of the hypergeometric differential equation.

The hypergeometric function contains many important functions as special cases, most notably the exponential and trigonometric functions. In fact, there are a large number of functions that can be written as a hypergeometric function.

Definition

The hypergeometric function is defined by

Where the gamma function.

Examples

Where Yes ( z) is the Bessel function

With ( modified Bessel function)

( γ (a, z ) is the incomplete gamma function is )

Evidence

Let us prove now the first examples. The ez has already been proved in the input text as obvious, so we now take the cosine:

Here we used that Γ (x 1 ) = xΓ (x ) and thus Γ (3/2 ) = 1/2 · Γ (1/2 ), etc. As you can see, the terms Γ ( 1, cut / 2) all out; the remaining fractions can easily be combined into

Let's try to prove the Polynombeispiel for a = 1:

Since the gamma function has negative values ​​at integer singularities, here must be made strictly limits. In the limit, then go to 0 because the count at last, however, the denominator becomes infinite. The first two links go counter as the denominator in the fraction tends to infinity. Since in each case are simple poles, but the quotients have finite limits: and. What remains is the finite polynomial 1 z This procedure is always the same, if you want to write polynomials as hypergeometric function.

Hypergeometric differential equation

The hypergeometric function is a solution of the hypergeometric differential equation

Leonhard Euler was a integral representation for the solution of the hypergeometric differential equation:

Each differential equation with three regular singular points can be transferred by transformation of the variables in the hypergeometric differential equation.

Further generalizations

The hypergeometric function can be further generalized by introducing pre-factors of k, and thus further increases the complexity of the function. Only the sign of k would modify two other indices necessary:

Are these pre-factors not necessary an integer, we obtain the generalization of the Fox -Wright functions.