The upper incomplete gamma function for some values of s: 0 (blue), 1 (red), 2 (green), 3 (orange), 4 (purple).
Plot of the regularized incomplete gamma function Q(2,z) in the complex plane from Γ(s,x)=∫x∞ts−1e−tdt,
and the lower incomplete gamma function is defined as
s is a complex number with positive real part.
Properties
editBy integration by parts, we find the recurrence relations Γ(s)=∫0∞ts−1e−tdt, we have x, can be developed into holomorphic functions, with respect both to x and γ(s,x)=∑k=0∞xse−xxks(s+1)⋯(s+k)=xsΓ(s)e−x∑k=0∞xkΓ(s+k+1). Given the rapid growth in absolute value of Γ(z + k) when s and γ∗, s) and C × C by Hartogs' theorem. Hence, the following decomposition s. It follows from the properties of z = 0 or 2πi only, which renders it multi-valued. Functions involving the complex logarithm typically inherit this property. Among these are the complex power, and, since ''s''</sup>"}},"i":0}}]}'/>z appears in its decomposition, the C × C called Riemann surface. While this removes multi-valuedness, one has to know the theory behind it;
The following set of rules can be used to interpret formulas in this section correctly. If not mentioned otherwise, the following is assumed:
Sectors
editSectors in D consists of all complex α − δ < arg z < α + δ with some α can be arbitrarily chosen and is not specified then. If C, with the exception of a half-line originating at α is silently taken to be 0, which centers the sector around the positive real axis.
Branches
editIn particular, a single-valued and holomorphic logarithm exists on any such sector D having its imaginary part bound to the range D (or α yields a different set of correlated branches on the same set |α| < δ, the branches are called principal, because they equal their real analogues on the positive real axis. Note: In many applications and texts, formulas hold only for principal branches.
Relation between branches
editThe values of different branches of both the complex power function and the lower incomplete gamma function can be derived from each other by multiplication of z = 0 asymptotically like: y and (x, y) → (0, s). This seems to justify setting s is positive, and (b) values ''v''</sup>"}},"i":0}}]}'/>u are taken from just a finite set of branches, they are guaranteed to converge to zero as γ(b) is naturally fulfilled, so there 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. In particular, the recurrence relation and ''s''−1</sup> ''e''<sup>−''z''</sup>"}},"i":0}}]}'/>∂γ(s, z)/∂z = z e are preserved on corresponding branches.
Integral representation
editThe last relation tells us, that, for fixed z e. Consequently, for any complex s is positive, then the limit γ 0. "}}'>γ(s,z)=∫0zts−1e−tdt,ℜ(s)>0.
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 z → +∞edit
Real values
editGiven the integral representation of a principal branch of x: 1 ≤ Re(s) ≤ 2 and |zs|=|z|ℜse−ℑsargz has been used in the middle. Since the final integral becomes arbitrarily small if only x → ∞ on the strip γ(s, x) = (s − 1) γ(s − 1, x) − x e and noting, that lim ''n''</sup> ''e''<sup>−''x''</sup> = 0"}},"i":0}}]}'/>x e = 0 for γ(s, x) converges outside the strip, too, towards a function obeying the recurrence relation of the Γ-function. It follows x real and |arg z| < δ < π/2 with some fixed γ be the principal branch on this sector, and look at |γ(s,|u|)−γ(s,u)|≤∫u|u||zs−1e−z|dz=∫u|u||z|ℜs−1e−ℑsargze−ℜzdz, where we made use of the integral representation of R = |u| around 0 connecting ≤R|argu|Rℜs−1eℑs|argu|e−Rcosargu≤δRℜseℑsδe−Rcosδ=M(Rcosδ)ℜse−Rcosδ where −Re ''s''</sup> ''e''<sup>Im ''sδ''</sup>"}},"i":0}}]}'/>M = δ(cos δ) e is a constant independent of x e for large ∞. In total we now have: 0 < ε < π/2 is arbitrarily small, but fixed, and z for fixed, positive integer s not an integer, with a branch point at z ≠ 0, with simple poles at non-positive integers s.
Upper incomplete gamma function
editAs for the upper incomplete gamma function, a holomorphic extension, with respect to Γ(s,z)=Γ(s)−γ(s,z) at points Γ, but a restriction to principal values only yields the single-valued principal branch of s → 0, fills in the missing values. Complex analysis guarantees holomorphicity, because γ∗ at γ, one obtains (assume γ(s,x)=∫0xts−1e−tdt=∫0x∑k=0∞(−1)kts+k−1k!dt=∑k=0∞(−1)kxs+kk!(s+k)=xs∑k=0∞(−x)kk!(s+k) or x (and all complex s → 0: Γ(0,z)=lims→0(Γ(s)−1s−(γ(s,z)−1s))=−γ−ln(z)−∑k=1∞(−z)kk(k!) is the limiting function to the upper incomplete gamma function as E1(z).
By way of the recurrence relation, values of Γ(−n,z)=1n!(e−zzn∑k=0n−1(−1)k(n−k−1)!zk+(−1)nΓ(0,z)) so the upper incomplete gamma function proves to exist and be holomorphic, with respect both to s and z for fixed, positive integral s non zero and not a positive integer, with a branch point at s with positive real part and s < 0!);
Here, erf is the error function, and γ(s,x)xs→1s as x→0 and Γ(s, x) ~ −x / s is on the order of min{''s'' + 1, 0}</sup>)"}},"i":0}}]}'/>O(x) if s = −1),
An alternative expansion is Connection with Kummer's confluent hypergeometric functionedit