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Multiplicative integral of gamma function is $$\int \Gamma(x)^{dx}=C \frac{e^{\psi^{(-2)}(x)}}{\Gamma(x)}$$e^{\psi^{(-2)}(x)}$$ if to use the popular generalization of polygamma function $\psi^{(p)}(z)$ put forward by Grossman in 1976. If to use a more modern "balanced" generalization $\psi(p,x)$ by Espinoza and Moll of 2004, the integral will be as follows: $$\int \Gamma(x)^{dx}=C e^{\psi(-2,x)-\frac e^{\psi(-2,x)+\frac {x}{2}\ln 2\pi}$$ |
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Multiplicative integral of gamma function is $$\int \Gamma(x)^{dx}=C \frac{e^{\psi^{(-2)}(x)}}{\Gamma(x)}$$ if to use the popular generalization of polygamma function $\psi^{(p)}(z)$ put forward by Groissman Grossman in 1976. If to use a more modern "balanced" generalization $\psi(p,x)$ by Espinoza and Moll of 2004, the integral will be as follows: $$\int \Gamma(x)^{dx}=C e^{\psi(-2,x)-\frac {x}{2}\ln 2\pi}$$ |
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Multiplicative integral of gamma function is $$\int \Gamma(x)^{dx}=C \frac{e^{\psi^{(-2)}(x)}}{\Gamma(x)}$$ if to use the popular generalization of polygamma function $\psi^{(p)}(z)$ put forward by Groissman in 1976. If to use a more modern "balanced" generalization $\psi(p,x)$ by Espinoza and Moll of 2004, the integral will be as follows: $$\int \Gamma(x)^{dx}=C e^{\psi(-2,x)-\frac {x}{2}\ln 2\pi}$$ |
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