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For $c=1-a-b$ with positive real parts $\mathrm{Re}a,\mathrm{Re}b,\mathrm{Re}c$, the integral converges at its two finite singularities and at infinity, and equalsTo prove it, use Schwinger parameterization with exponents $1-a$ and $1-b$. Integrating the resulting planar Gaussian integral gives $\pi /(\lambda +\rho )$ times $\exp [-\lambda \rho /(\lambda +\rho )]$. Set $q=\lambda +\rho$ and $x=\lambda /q$; the $q$ integral is a gamma function and the $x$ integral is $B(a,b)$, giving the stated ratio. This domain justifies the interchanges; elsewhere the answer is interpreted by analytic continuation. The string measure $d^{2}z=2dxdy$ multiplies the answer by two.

If the first two phase derivatives vanish at an endpoint but the third does not, an oscillatory contribution is concentrated on a width proportional to $N^{-1/3}$. For a smooth nonzero endpoint oscillatory integral amplitude, rescaling reduces the leading contribution to a cubic oscillatory integral. Its constant follows from the Gamma function Fourier integral. This is a degenerate form of the stationary phase method, and a quadratic formula is insufficient.

For $z$ away from the poles, the gamma function satisfiesIts finite product is $m!(m+1{)}^z/[z(z+1)\cdots (z+m)]$. Splitting even and odd factors gives the gamma duplication formula.

For $t\simeq -as$, $u\simeq -(1-a)s$ with $0<a<1$, the smooth high-energy envelope satisfies $\log |A|\simeq -2s[-a\log a-(1-a)\log (1-a)]$. The Gamma function asymptotics cancel the leading $s\log s$ terms, leaving exponential suppression. Exact real-axis resonance poles must be excluded or treated by analytic continuation.

On $z>0$, consider $p_t+a{p}_z=D(z{p}_z{)}_z$ with $a,D>0$, zero scalar flux $ap-Dz{p}_z$ at the endpoints for $t>0$, and unit initial impulse at the origin. Put $r=a/D$ and $\eta =z/(Dt)$. The normalized similarity solution isThe ansatz $p=(Dt{)}^{-1}f(\eta )$ gives $[\eta f^{\prime }+(\eta -r)f{]}^{\prime }=0$. Endpoint decay sets this integrated constant to zero and hence $f\propto {\eta }^r{e}^{-\eta }$; the gamma function normalizes its integral to one. Its scale is $Dt$, so it converges weakly to a unit impulse as $t\downarrow 0$. Its gamma distribution shape is $1+r$, its maximum occurs at $z=at$, and its expected value is $(a+D)t$. For width proportional to $z$, the associated horizontally averaged plume concentration is proportional to $z^{r-1}{e}^{-z/(Dt)}$: its interior maximum is $(a-D)t$ when $a>D$, its supremum occurs at the origin when $a=D$, and it is singular there when $0<a<D$. The physical finite source regularizes this ideal-origin behaviour.

For $a,s>0$, the change of variable $u=at$ in the defining integral for the Gamma function givesIn particular, $\Gamma (1/2)=\sqrt{\pi }$ implies

With a mostly-plus metric signature, define $f_d=(1/i)\int d^{d}k/[(2\pi {)}^d(k^2-iϵ)((P-k{)}^2-iϵ)]$. Wick rotation and a Feynman parameter give $f_d=\Gamma (2-d/2)(4\pi {)}^{-d/2}{\int }_0^1[x(1-x)P^2{]}^{d/2-2}dx$. At nonzero Euclidean $P^2$, the Gamma function supplies the pole. Exceptional massless momentum can introduce infrared ambiguity.

For a nonprincipal Dirichlet character modulo $N$, setThe numerator vanishes at $t=0$, so $H_{\chi }$ is analytic there and exponentially decreasing as $t\to \infty$. Initially for $\mathrm{Re}s>1$, $\Gamma (s)L(\chi ,s)={\int }_0^{\infty }{H}_{\chi }(t)t^{s-1}dt$. This integral already extends holomorphically to $\mathrm{Re}s>0$. Subtracting finite Taylor polynomials near zero, as in meromorphic continuation of a Mellin transform from an asymptotic expansion, continues it with possible simple poles only at nonpositive integers. Zeros of the reciprocal Gamma function cancel these, proving that $L(\chi ,s)$ is an entire function. This does not require a primitive Dirichlet character.

The solution of the Modified Bessel differential equation with power series $I_{\nu }(z)={\sum }_{j\ge 0}(z/2{)}^{2j+\nu }/[j!\Gamma (j+\nu +1)]$, interpreted through the reciprocal gamma function at its zeros. For fixed real order it grows exponentially along the positive real axis.