# double gamma function

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##### 1: 5.17 Barnes’ $G$-Function (Double Gamma Function)
###### §5.17 Barnes’ $G$-Function (DoubleGammaFunction)
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5.17.3 $G\left(z+1\right)=(2\pi)^{z/2}\exp\left(-\tfrac{1}{2}z(z+1)-\tfrac{1}{2}\gamma z% ^{2}\right)\*\prod_{k=1}^{\infty}\left(\left(1+\frac{z}{k}\right)^{k}\exp\left% (-z+\frac{z^{2}}{2k}\right)\right).$
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5.17.4 $\operatorname{Ln}G\left(z+1\right)=\tfrac{1}{2}z\ln\left(2\pi\right)-\tfrac{1}% {2}z(z+1)+z\operatorname{Ln}\Gamma\left(z+1\right)-\int_{0}^{z}\operatorname{% Ln}\Gamma\left(t+1\right)\,\mathrm{d}t.$
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5.17.5 $\operatorname{Ln}G\left(z+1\right)\sim\tfrac{1}{4}z^{2}+z\operatorname{Ln}% \Gamma\left(z+1\right)-\left(\tfrac{1}{2}z(z+1)+\tfrac{1}{12}\right)\ln z-\ln A% +\sum_{k=1}^{\infty}\frac{B_{2k+2}}{2k(2k+1)(2k+2)z^{2k}}.$
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##### 3: Bibliography F
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• FDLIBM (free C library)
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• C. Ferreira and J. L. López (2001) An asymptotic expansion of the double gamma function. J. Approx. Theory 111 (2), pp. 298–314.
• ##### 4: Bibliography M
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• C. Mortici (2013b) Further improvements of some double inequalities for bounding the gamma function. Math. Comput. Modelling 57 (5-6), pp. 1360–1363.
• ##### 5: 5.4 Special Values and Extrema
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5.4.2 $n!!=\begin{cases}2^{\frac{1}{2}n}\Gamma\left(\frac{1}{2}n+1\right),&n\text{ % even},\\ \pi^{-\frac{1}{2}}2^{\frac{1}{2}n+\frac{1}{2}}\Gamma\left(\frac{1}{2}n+1\right% ),&n\text{ odd}.\end{cases}$
##### 6: Bibliography Q
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• F. Qi and J. Mei (1999) Some inequalities of the incomplete gamma and related functions. Z. Anal. Anwendungen 18 (3), pp. 793–799.
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• F. Qi (2008) A new lower bound in the second Kershaw’s double inequality. J. Comput. Appl. Math. 214 (2), pp. 610–616.
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• W.-Y. Qiu and R. Wong (2000) Uniform asymptotic expansions of a double integral: Coalescence of two stationary points. Proc. Roy. Soc. London Ser. A 456, pp. 407–431.
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• C. K. Qu and R. Wong (1999) “Best possible” upper and lower bounds for the zeros of the Bessel function $J_{\nu}(x)$ . Trans. Amer. Math. Soc. 351 (7), pp. 2833–2859.
• ##### 7: 8.13 Zeros
###### §8.13 Zeros
βΊThe function $\gamma^{*}\left(a,x\right)$ has no real zeros for $a\geq 0$. … βΊWhen $-5\leq a\leq 4$ the behavior of the $x$-zeros as functions of $a$ can be seen by taking the slice $\gamma^{*}\left(a,x\right)=0$ of the surface depicted in Figure 8.3.6. … βΊAs $x$ increases the positive zeros coalesce to form a double zero at ($a_{n}^{*},x_{n}^{*}$). The values of the first six double zeros are given to 5D in Table 8.13.1. …
##### 8: 21.5 Modular Transformations
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21.5.5 $\boldsymbol{{\Gamma}}=\begin{bmatrix}\mathbf{A}&\boldsymbol{{0}}_{g}\\ \boldsymbol{{0}}_{g}&[\mathbf{A}^{-1}]^{\mathrm{T}}\end{bmatrix}\Rightarrow% \theta\left(\mathbf{A}\mathbf{z}\middle|\mathbf{A}\boldsymbol{{\Omega}}\mathbf% {A}^{\mathrm{T}}\right)=\theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}% \right).$
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21.5.6 $\boldsymbol{{\Gamma}}=\begin{bmatrix}\mathbf{I}_{g}&\mathbf{B}\\ \boldsymbol{{0}}_{g}&\mathbf{I}_{g}\end{bmatrix}\Rightarrow\theta\left(\mathbf% {z}\middle|\boldsymbol{{\Omega}}+\mathbf{B}\right)=\theta\left(\mathbf{z}% \middle|\boldsymbol{{\Omega}}\right).$
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21.5.7 $\boldsymbol{{\Gamma}}=\begin{bmatrix}\mathbf{I}_{g}&\mathbf{B}\\ \boldsymbol{{0}}_{g}&\mathbf{I}_{g}\end{bmatrix}\Rightarrow\theta\left(\mathbf% {z}\middle|\boldsymbol{{\Omega}}+\mathbf{B}\right)=\theta\left(\mathbf{z}+% \tfrac{1}{2}\operatorname{diag}\mathbf{B}\middle|\boldsymbol{{\Omega}}\right).$
##### 9: 16.15 Integral Representations and Integrals
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16.15.3 ${F_{3}}\left(\alpha,\alpha^{\prime};\beta,\beta^{\prime};\gamma;x,y\right)=% \frac{\Gamma\left(\gamma\right)}{\Gamma\left(\beta\right)\Gamma\left(\beta^{% \prime}\right)\Gamma\left(\gamma-\beta-\beta^{\prime}\right)}\iint_{\Delta}% \frac{u^{\beta-1}v^{\beta^{\prime}-1}(1-u-v)^{\gamma-\beta-\beta^{\prime}-1}}{% (1-ux)^{\alpha}(1-vy)^{\alpha^{\prime}}}\,\mathrm{d}u\,\mathrm{d}v,$ $\Re\left(\gamma-\beta-\beta^{\prime}\right)>0$, $\Re\beta>0$, $\Re\beta^{\prime}>0$,
##### 10: 15.6 Integral Representations
###### §15.6 Integral Representations
βΊIn (15.6.1) all functions in the integrand assume their principal values. … βΊIn (15.6.6) the integration contour separates the poles of $\Gamma\left(a+t\right)$ and $\Gamma\left(b+t\right)$ from those of $\Gamma\left(-t\right)$, and $(-z)^{t}$ has its principal value. βΊIn (15.6.7) the integration contour separates the poles of $\Gamma\left(a+t\right)$ and $\Gamma\left(b+t\right)$ from those of $\Gamma\left(c-a-b-t\right)$ and $\Gamma\left(-t\right)$, and $(1-z)^{t}$ has its principal value. βΊIn each of (15.6.8) and (15.6.9) all functions in the integrand assume their principal values. …