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Lower Incomplete Gamma Function

Holomorphic Extension

Repeated application of the recurrence relation for the lower incomplete gamma function leads to the power series expansion: [1]

$\gamma (s,x)=\sum _{k=0}^{\infty }{\frac {x^{s}e^{-x}x^{k}}{s(s+1)...(s+k)}}=x^{s}\,\Gamma (s)\,e^{-x}\sum _{k=0}^{\infty }{\frac {x^{k}}{\Gamma (s+k+1)}}$

Given the rapid growth in absolute value of $\Gamma (z+k)$ 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. By a theorem of Weierstraß, the limiting function, sometimes denoted as $\gamma ^{*}$,

$\gamma ^{*}(s,z):=e^{-z}\sum _{k=0}^{\infty }{\frac {z^{k}}{\Gamma (s+k+1)}}$ [2]

is entire with respect to both z (for fixed s) and s (for fixed z) [3], and, thus, holomorphic on ℂ×ℂ by Hartog's theorem[4]. Hence, the following decomposition

$\gamma (s,z)=z^{s}\,\Gamma (s)\,\gamma ^{*}(s,z)$ [5],

extends the real lower incomplete gamma function as a holomorphic function, both jointly and separately in z and s. It follows from the properties of z^s and the Γ-function, that the first two factors capture the singularities of γ (at z = 0 or s a non-positive integer), whereas the last factor contributes to its zeros.

Branches

The complex logarithm log z = ln |z| + i arg z is determined up to a multiple of 2πi only, which renders it multi-valued. Functions involving the complex logarithm typically inherit this property. Among these are the complex power, and, since z^s appears in its decomposition, the γ-function, too.

The indeterminacy of multi-valued functions introduces complications, since it must be stated how to select a value. Among the strategies to handle this are:

• always explicitly state what value to select, which is cumbersome;
• (the most general way) replace the domain ℂ of multi-valued functions by a suitable manifold in ℂ×ℂ called Riemann surface. While this removes multi-valuedness, one has to know the theory behind it[6];
• restrict the domain such that a multi-valued function decomposes into separate single-valued branches, which can be handled individually.

The following set of rules can be used to interpret formulas in this section correctly. If not mentioned otherwise, the following is assumed:

(a) Sectors in ℂ having their vertex at z = 0 often prove to be appropriate domains for complex expressions. A sector D consists of all complex z fulfilling z ≠ 0 and α - δ < arg z < α + δ with some α and 0 < δ ≤ π. Often, α need not be specified, and can be arbitrarily choosen. 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, the so-called branch cut. Note: In many applications and texts, α is silently taken to be 0, which centers the sector around the positive real axis.

(b) In particular, a single-valued and holomorphic logarithm exists on any such sector D having its imaginary part bound to the range (α - δ, α + δ). Based on such a restricted logarithm, z^s 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. If |α| < δ, the branches are called principal, because they equal their real analogons on the positive real axis. Note: In many applications and texts, formulas hold only for principal branches.

(c) Finally, the expression e^s shall always denote the exponential function, which is the restriction of a principal branch of z^s to z = e.

The values of different branches of both the complex power function and the lower incomplete gamma function can be derived from each other by multiplication by $e^{s*2k\pi i}$[7], k a suitable integer.

Behavior near Branch Point

The decomposition above further shows, that γ behaves near z = 0 asymptotically like:

$\gamma (s,z)\asymp z^{s}\,\Gamma (s)\,\gamma ^{*}(s,0)=z^{s}\,\Gamma (s)/\Gamma (s+1)=z^{s}/s$

For positive real x, y and s, x^y/y → 0, when (x, y) → (0, s). This seems to justify setting γ(s, 0) = 0 for real s > 0. However, matters are somewhat different in the complex realm. Only if (a) the real part of s is positive, and (b) values u^v 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). On a single branch of γ (b) is naturally fulfilled, so there γ(s, 0) = 0 for s with positive real part is a continuous limit. Also note that such a continuation is by no means an analytic one.

Algebraic Relations

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 [8], stating that equations between holomorphic functions valid on a real interval, hold everywhere. In particular, the recurrence relation [9] and ∂γ(s,z)/∂z = z^s-1 e^-z [10] are preserved on corresponding branches.

Integral Representation

The last relation tells us, that, for fixed s, γ is a primitive or antiderivative of the holomorphic function z^s-1 e^-z. Consequently [11], for any complex u, v ≠ 0,

$\int _{u}^{v}t^{s-1}\,e^{-t}\,{\rm {d}}t=\gamma (s,v)-\gamma (s,u)$

holds, as long as the path of integration is entirely contained in the domain of a branch of the integrand. If, additionally, the real part of s is positive, then the limit γ(s, u) → 0 for u → 0 applies, finally arriving at the complex integral definition of γ

$\gamma (s,z)=\int _{0}^{z}t^{s-1}\,e^{-t}\,{\rm {d}}t,\,\Re s>0.$[12]

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.

Limit for z → ∞

s, x real

Given the integral representation of a principal branch of γ, the following equation holds for all positive real s:[13]

$\Gamma (s)=\int _{0}^{\infty }t^{s-1}\,e^{-t}\,{\rm {d}}t=\lim _{x\rightarrow \infty }\gamma (s,x)$

s complex

This result extends to complex s. Assume first 1 ≤ Re(s) ≤ 2 and 1 < a < b. Then

$|\gamma (s,b)-\gamma (s,a)|\leq \int _{a}^{b}|t^{s-1}|e^{-t}\,{\rm {d}}t=\int _{a}^{b}t^{\Re s-1}e^{-t}\,{\rm {d}}t\leq \int _{a}^{b}te^{-t}\,{\rm {d}}t$

where

$|z^{s}|=|z|^{\Re s}\,e^{-\Im s\arg z}$[14]

has been used in the middle. Since the final integral can be made arbitrarily small if only a is choosen big enough, γ(s, x) converges uniformly for x → ∞ on the strip 1 ≤ Re(s) ≤ 2 towards a holomorphic function , which must be Γ(s) because of the identity theorem [[15]. Taking the limit in the recurrence relation γ(s,x) = (s-1)γ(s-1,x) - x^s-1 e^-x and noting, that lim xⁿ e^-x = 0 for x → ∞ and all n, shows, that γ(s,x) converges outside the strip, too, towards a function obeying the recurrence relation of the Γ-function. It follows

$\Gamma (s)=\lim _{x\rightarrow \infty }\gamma (s,x)$

for all complex s not a non-positive integer, x real and γ principal.

complex convergence

Now let u be from the sector |arg z| < δ < π/2 with some fixed δ (α = 0), γ be the principal branch on this sector, and look at

$\Gamma (s)-\gamma (s,u)=\Gamma (s)-\gamma (s,|u|)+\gamma (s,|u|)-\gamma (s,u).$

As shown above, the first difference can be made arbitrarily small, if |u| is sufficiently large. The second difference allows for following estimation:

$|\gamma (s,|u|)-\gamma (s,u)|\leq \int _{u}^{|u|}|z^{s-1}e^{-z}|\,{\rm {d}}z=\int _{u}^{|u|}|z|^{\Re s-1}\,e^{-\Im s\,\arg z}\,e^{-\Re z}\,{\rm {d}}z$

where we made use of the integral representation of γ and the formula about |z^s| above. If we integrate along the arc with radius R = |u| around 0 connecting u and |u|, then the last integral is

$\leq R|\arg u|\,R^{\Re s-1}\,e^{\Im s\,|\arg u|}\,e^{-R\cos \arg u}\leq \delta \,R^{\Re s}\,e^{\Im s\,\delta }\,e^{-R\cos \delta }=M\,(R\,\cos \delta )^{\Re s}\,e^{-R\cos \delta }$

where M = δ (cos δ)^{-Re s} e^{Im s δ} is a constant independent of u or R. Again referring to the behavior of xⁿ e^-x for large x, we see that the last expression approaches 0 as R increases towards ∞. In total we now have:

$\Gamma (s)=\lim _{|z|\rightarrow \infty }\gamma (s,z),\quad |\arg z|<\pi /2-\epsilon$

if |arg z| < π/2 - ε for any 0 < ε < π/2 arbitrarily small, but fixed, and γ denoting the principal branch in this domain.

Overview

$\gamma (s,z)$ is:

• entire in z for fixed, positive integral s;
• multi-valued holomorphic in z for fixed s not an integer, with a branch point at z = 0;
• on each branch meromorphic in s for fixed z ≠ 0, with simple poles at non-positive integers s.

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