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The Gauss series disk and a continued logarithmic branch have different domains

Statement

For a=b=1 and c=2, the Gauss-series branch in Fresnel, error, and Gauss hypergeometric series is defined for ∣z∣<1 and there equals 2F1(1,1;2;z)=∑k=0∞zkk+1=−Log⁡(1−z)z, with value 1 at z=0. The right side has a principal analytic continuation to C∖[1,∞), which includes z=−2; the series evaluator correctly returns domain at z=−2 because this input lies outside the defined power-series branch. This example distinguishes the series branch's disk from the continuation branch's slit domain.

Facts & Assumptions

Given: The exact parameters a=b=1, c=2.

[F1]

Fresnel, error, and Gauss hypergeometric series: the Gauss branch is the stated power series on ∣z∣<1 with c not a nonpositive integer.

[F2]

The principal logarithm is the normalised holomorphic branch on the slit plane: Log⁡′(w)=1/w away from the nonpositive real axis.

[F3]

The second fundamental theorem: if G is differentiable on [a,b] with G′=f and f is integrable, then ∫abf=G(b)−G(a): a continuously differentiable complex path satisfies the fundamental theorem of calculus by applying the real theorem to its real and imaginary parts.

[F4]

A continuous function on [a,b] is Riemann integrable, by Heine-Cantor and Riemann's criterion and Inside its radius a real power series may be integrated term by term on every closed subinterval: a continuous integrand on [0,1] is integrable and a real power series may be integrated termwise strictly inside its radius; apply the latter to both real and imaginary component power series along the segment.

[F5]

Finite certified power-series evaluation for Fresnel, erf, and Gauss functions: its Gauss-series branch returns domain unless c avoids the excluded integers and ∣z∣<1.

Proof

1.1F1F2F3F4algebra

For a=b=1,c=2, the Pochhammer products give (1)k=k! and (2)k=(k+1)!, so each coefficient is 1/(k+1). For ∣z∣<1, the geometric series converges uniformly along the segment sz (0≤s≤1). Parameterizing the path and integrating the real and imaginary component power series termwise by [F4] gives ∑k=0∞zk+1k+1=∫01z1−sz ds=−Log⁡(1−z), where [F2], [F3], and [F4] justify the path integral identity since Re⁡(1−sz)>0 along the segment. Dividing by z gives the formula, and the series gives value 1 at z=0.

2.1F2step 1.1algebra

The principal logarithm is holomorphic when 1−z avoids the nonpositive real axis, that is, when z∉[1,∞). The quotient has a removable value at z=0 by step 1.1, so it defines the stated continuation domain, which contains −2.

3.1F1F5step 2.1∎

Since ∣−2∣>1, the defined Gauss-series branch fails its domain check and [F5] returns domain; this does not contradict the continued formula in step 2.1, which is a separately declared branch.

Depends on

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