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Tag Archives: Digamma
Integral representation of the digamma function using Abel–Plana formula
$$ \int^\infty_0 \frac{2x}{(x^2+z^2)e^{2\pi x}1}\,dx =\log(z)\psi(z)\frac{1}{2z}$$ $$\textit{proof}$$ Use Abel–Plana formula $$\sum_{n=0}^\infty f(n) = \int^\infty_0f(x)\,dx+\frac{f(0)}{2} +i\int^\infty_0 \frac{f(ix)f(ix)}{e^{2\pi x}1}\,dx$$ Let $$f(x) = \frac{1}{z+x}$$ Note that $$i(f(ix) f(ix))= \frac{i}{z+ix}\frac{i}{zix} = \frac{2x}{z^2+x^2}$$ By integration we have $$ \int^\infty_0 \frac{2x}{(x^2+z^2)e^{2\pi x}1}\,dx =\lim_{N\to \infty}\sum_{n=0}^N \frac{1}{z+n}\int^N_0 \frac{1}{x+z}\,dx\frac{1}{2z}$$ The … Continue reading
Posted in Digamma function
Tagged Abel–Plana, Digamma, formula, function, Integral, proof, representation
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Relation between polygamma and Hurwitz zeta function proof
\( \forall \,\, n\geq 1 \) $$\psi_{n}(z) \, = \, (1)^{n+1}n!\,\zeta(n+1,z)$$ $$\textit{proof}$$ Use the series representation of the digamma $$\psi_{0}(z) = \gamma\frac{1}{z}+ \sum_{n=1}^\infty\frac{z}{n(n+z)}$$ This can be written as the following $$\psi_{0}(z) = \gamma + \sum_{k=0}^\infty\frac{1}{k+1}\frac{1}{k+z}$$ By differentiating with respect to … Continue reading
The most ugly looking integral
Prove the following $$I= \log \left\{\frac{\Gamma(b+c+1) \Gamma(c+a+1)\Gamma(a+b+1)}{\Gamma(a+1) \Gamma(b+1) \Gamma(c+1) \Gamma(a+b+c+1)} \right\}$$ where $$I = \int_0^1 \frac{(1x^a)(1x^b)(1x^c)}{(1x)(\log x)}dx$$ $$\textit{proof}$$ First note that since there is a log in the denominator that gives as an idea to use differentiation under … Continue reading
Posted in Digamma function, Gamma function
Tagged difficult, Digamma, Gamma, Hard, Integral, proof
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Digamma fourth integral representation proof
$$\psi(z) = \log(z) \frac{1}{2z}2\int^\infty_0 \frac{t}{(t^2+z^2)(e^{2\pi}1)}dt\,\,\,\,;\text{ Re}z>0$$ We prove that $$2\int^\infty_0 \frac{t}{(t^2+z^2)(e^{2\pi}1)}dt= \log(z) \frac{1}{2z} \psi(z)$$ First note that $$\frac{2}{e^{2\pi t}1} =\coth(\pi t)1 $$ Also note that $$\coth(\pi t) = \frac{1}{\pi t}+\frac{2t}{\pi}\sum_{k=1}^\infty\frac{1}{k^2+t^2}$$ Hence we conclude that $$\frac{2t}{e^{2\pi t}1} =\frac{1}{\pi}t+\frac{2t^2}{\pi}\sum_{k=1}^\infty\frac{1}{k^2+t^2} $$ Substitute the … Continue reading
Digamma third integral representation proof
$$\psi \left(a\right)=\int^{\infty}_0 \, \frac{e^{t}}{t}\frac{e^{\left(a t\right)}}{1e^{t}}\, dt$$ $$\textit{proof}$$ Let \( e^{t}=x \) $$\int^{1}_0 \, \frac{1}{\log(x)}\frac{x^{a1}}{1x}\, dx$$ By adding and subtracting 1 $$\int^{1}_0 \, \frac{1}{\log(x)}+\frac{1}{1x}\, dx+\int^1_0\frac{1x^{a1}}{1x}\, dx$$ Using the second integral representation $$\int^{1}_0 \, \frac{1}{\log(x)}+\frac{1}{1x}\, dx+\gamma+\psi(a)$$ We can prove that $$\int^{1}_0 \, … Continue reading
Second integral representation of digamma proof
$$\psi(s+1)\,=\, \gamma \,+\, \int^{1}_{0}\frac{1x^s}{1x}\,dx$$ $$\textit{proof}$$ This can be done by noting that $$\psi(s+1) = \gamma +\sum_{n=1}^\infty\frac{s}{n(n+s)}$$ It is left as an exercise to prove that $$\sum_{n=1}^\infty\frac{s}{n(n+s)} = \int^{1}_{0}\frac{1x^s}{1x}\,dx$$
First integral representation of digamma proof
$$ \psi(a) = \int^{\infty}_0 \frac{e^{z}(1+z)^{a}}{z}\,dz $$ $$\textit{proof}$$ We begin with the double integral $$\int^{\infty}_0 \int^t_1 \,e^{xz}\,dx\,dz=\int^{\infty}_{0}\frac{e^{z}e^{tz}}{z}\, dz $$ Using fubini theorem we also have $$ \int^t_1 \int^{\infty}_0\,e^{xz}\,dz \,dx = \int^t_1 \frac{1}{x}\,dx = \log t $$ Hence we have the following … Continue reading
Digamma difference formula proof
$$\psi(1x)\psi(x)=\pi \cot(\pi x) $$ $$\textit{proof}$$ We know by the reflection formula that $$\Gamma(x)\Gamma(1x)=\pi \csc(\pi x) $$ Now differentiate both sides $$\psi(x)\Gamma(x)\Gamma(1x)\psi(1x)\Gamma(x)\Gamma(1x)=\pi^2 \csc(\pi x)\,\cot(\pi x) $$ Which can be simplified $$\Gamma(x)\Gamma(1x)\left(\psi(1x)\psi(x)\right)=\pi^2 \csc(\pi x)\,\cot(\pi x) $$ Further simplifications using ERF results in $$ … Continue reading