Pochhammer double-loop contour

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1: 31.9 Orthogonality

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31.9.2 $${\int }_{\zeta }^{(1+,0+,1-,0-)}{t}^{\gamma -1}{(1-t)}^{\delta -1}{(t-a)}^{ϵ-1}{w}_m(t)w_k(t)dt={\delta }_{m,k}{\theta }_m.$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\int$: integral, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $\zeta$: point and ${\theta }_m$: normalization constant

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http://dlmf.nist.gov/31.9.E2

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Annotations for §31.9(i), §31.9 and Ch.31

… ►The integration path is called a Pochhammer double-loop contour (compare Figure 5.12.3). … ►and the integration paths ${ℒ}_1$, ${ℒ}_2$ are Pochhammer double-loop contours encircling distinct pairs of singularities $\{0,1\}$, $\{0,a\}$, $\{1,a\}$. …

2: 15.6 Integral Representations

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Figure 15.6.1: $t$-plane. … Magnify

3: 13.4 Integral Representations

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Figure 13.4.1: Contour of integration in (13.4.11). … Magnify

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4: 31.10 Integral Equations and Representations

… ►where $\mathrm{\Re }\gamma >0$, $\mathrm{\Re }\delta >0$, and $C$ be the Pochhammer double-loop contour about 0 and 1 (as in §31.9(i)). …