Incomplete gamma function

This contrasts with the lower incomplete gamma function, which is defined as an integral from zero to a variable upper limit.

Similarly, the upper incomplete gamma function is defined as an integral from a variable lower limit to infinity.

The lower incomplete gamma and the upper incomplete gamma function, as defined above for real positive s and x, can be developed into holomorphic functions, with respect both to x and s, defined for almost all combinations of complex x and s.[1] Complex analysis shows how properties of the real incomplete gamma functions extend to their holomorphic counterparts.

Repeated application of the recurrence relation for the lower incomplete gamma function leads to the power series expansion: [2]

Given the rapid growth in absolute value of Γ(z + k) when k → ∞, and the fact that the reciprocal of Γ(z) is an entire function, the coefficients in the rightmost sum are well-defined, and locally the sum converges uniformly for all complex s and x.

The indeterminacy of multi-valued functions introduces complications, since it must be stated how to select a value.

Strategies to handle this are: The following set of rules can be used to interpret formulas in this section correctly.

If not mentioned otherwise, the following is assumed: Sectors in C having their vertex at z = 0 often prove to be appropriate domains for complex expressions.

If δ is not given, it is assumed to be π, and the sector is in fact the whole plane C, with the exception of a half-line originating at z = 0 and pointing into the direction of −α, usually serving as a branch cut.

Note: In many applications and texts, α is silently taken to be 0, which centers the sector around the positive real axis.

Based on such a restricted logarithm, zs and the incomplete gamma functions in turn collapse to single-valued, holomorphic functions on D (or C×D), called branches of their multi-valued counterparts on D. Adding a multiple of 2π to α yields a different set of correlated branches on the same set D. However, in any given context here, α is assumed fixed and all branches involved are associated to it.

Note: In many applications and texts, formulas hold only for principal branches.

Only if (a) the real part of s is positive, and (b) values uv are taken from just a finite set of branches, they are guaranteed to converge to zero as (u, v) → (0, s), and so does γ(u, v).

All algebraic relations and differential equations observed by the real γ(s, z) hold for its holomorphic counterpart as well.

This is a consequence of the identity theorem, stating that equations between holomorphic functions valid on a real interval, hold everywhere.

The last relation tells us, that, for fixed s, γ is a primitive or antiderivative of the holomorphic function zs−1 e−z.

holds, as long as the path of integration is entirely contained in the domain of a branch of the integrand.

Any path of integration containing 0 only at its beginning, otherwise restricted to the domain of a branch of the integrand, is valid here, for example, the straight line connecting 0 and z.

Given the integral representation of a principal branch of γ, the following equation holds for all positive real s, x:[7]

if s is not a non-negative integer, 0 < ε < π/2 is arbitrarily small, but fixed, and γ denotes the principal branch on this domain.

When s is a non-positive integer in the above equation, neither part of the difference is defined, and a limiting process, here developed for s → 0, fills in the missing values.

so the upper incomplete gamma function proves to exist and be holomorphic, with respect both to z and s, for all s and z ≠ 0.

The lower gamma function can be evaluated using the power series expansion:[15]

For the actual computation of numerical values, Gauss's continued fraction provides a useful expansion:

This continued fraction converges for all complex z, provided only that s is not a negative integer.

In Python, the Scipy library provides implementations of incomplete gamma functions under scipy.special, however, it does not support negative values for the first argument.

is the cumulative distribution function for gamma random variables with shape parameter

This particular special case has internal closure properties of its own because it can be used to express all successive derivatives.

When combined with a computer algebra system, the exploitation of special functions provides a powerful method for solving definite integrals, in particular those encountered by practical engineering applications (see Symbolic integration for more details).

The lower and the upper incomplete gamma function are connected via the Fourier transform: