
Recent Posts
 Integrating a cosine log integral around a semicircle contour
 Creating Difficult integrals by the residue theorem
 Proving a trigonometric integral by integrating around an ellipse in the complex plain
 Integrating a fraction of exponential and trignometric using rectangular contour
 Integrating around a triangular contour for Fresnel integral
Recent Comments
 Ricardo on Integral representation of the digamma function using Abel–Plana formula
 Zaidalyafeai on Integral representation of the digamma function using Abel–Plana formula
 Ricardo on Integral representation of the digamma function using Abel–Plana formula
 Zaidalyafeai on Integral of arctan and log using contour integration
 tired on Integral of arctan and log using contour integration
Archives
Categories
Meta
Monthly Archives: January 2017
Relation between harmonic numbers and Stirling numbers of the first kind
Prove that $$\left[n\atop 2\right] = H_{n1}\Gamma(n)$$ By induction on \( n\) we have for \( n=2\) $$\left[2\atop 2\right] = H_{1}\times\Gamma(1) = 1$$ Assume that $$\left[k\atop 2\right] = H_{k1}\Gamma(k)$$ Then by the recurrence relation $$\left[k+1\atop 2\right] = k\left[k\atop 2\right] + \left[k\atop … Continue reading
Posted in Harmonic numbers, Striling numbers of first kind
Tagged Harmonic, number, numbers, proof, relation, stirling
Leave a comment
Stirling numbers of the first kind special values proof
Prove that $$\left[n\atop 1\right] = \Gamma(n)$$ Use the recurrence relation $$\left[n+1\atop k\right] = n\left[n\atop k\right] + \left[n\atop k1\right]$$ This implies that for \( k=1 \) $$\left[n+1\atop 1\right] = n\left[n\atop 1\right] + \left[n\atop0\right]$$ Now use that \( \left[n\atop 0\right] = 0 … Continue reading
Posted in Striling numbers of first kind
Tagged first, kind, numbers, proof, special, stirling, values
Leave a comment
Special values of the dilgoarithm function
Prove that $$\mathrm{Li}_2\left( \frac{\sqrt{5}1}{2} \right) = \frac{\pi^2}{10} – \log^2 \left( \frac{\sqrt{5}1}{2}\right)$$ $$\textit{proof}$$ Use the following functional equation $$\mathrm{Li}_2 \left( \frac{z}{z1} \right) + \frac{1}{2} \mathrm{Li}_2 (z^2) – \mathrm{Li}_2(z) = \frac{1}{2} \log^2 (1z) $$ These are proved here and here Now let … Continue reading
Posted in Dilogarithm, Polylogarithm
Leave a comment
Square difference formula for polylgoarithm proof
$$\mathrm{Li}_{\,n}(z) + \mathrm{Li}_{\,n}(z) = 2^{1n} \,\mathrm{Li}_{\,n}(z^2) $$ $$\textit{proof}$$ As usual we write the series representation of the LHS $$\sum_{k=1}^\infty \frac{z^k}{k^n}+\sum_{k=1}^\infty \frac{(z)^k}{k^n}$$ Listing the first few terms $$z+\frac{z^2}{2^n}+\frac{z^3}{3^n}+\cdots +\left(z+\frac{z^2}{2^n}\frac{z^3}{3^n}+\cdots \right) $$ The odd terms will cancel $$2\frac{z^2}{2^n}+2\frac{z^4}{4^n}+2\frac{z^6}{6^n}+\cdots $$ Take \( 2^{1n} … Continue reading
Posted in Dilogarithm, Polylogarithm
Tagged difference, dilogarithm, formula, polylgoarithm, proof, recurrence, square
1 Comment
Dilogarithm difference formula proof
$$\mathrm{Li}_2(z) + \mathrm{Li}_2 \left(\frac{z}{z1} \right) = – \frac{1}{2} \log^2 (1z) \,\,\,\, \, z<1$$ $$ \textit{proof} $$ Start by the following $$\mathrm{Li}_2 \left(\frac{z}{z1} \right) = \int^{\frac{z}{z1}}_0 \frac{ \log(1t)}{t}\, dt$$ Differentiate both sides with respect to $z$ $$\frac{d}{dz}\mathrm{Li}_2 \left(\frac{z}{z1} \right) = \frac{1}{(z1)^2}\left( \frac{ \log … Continue reading
Dilogarithm at 2
$$\mathrm{Li}_2\left(\frac{1}{2}\right)= \frac{\pi^2}{12}\frac{1}{2}\log^2 \left(\frac{1}{2}\right) $$ $$ proof $$ Using the duplication formula proved here $$\mathrm{Li}_2\left(z\right)+\mathrm{Li}_2(1z)\, = \frac{\pi^2}{6}\log(z) \log(1z) \,\, $$ We can easily deduce that for \( z=\frac{1}{2}\) $$2\mathrm{Li}_2\left(\frac{1}{2}\right)= \frac{\pi^2}{6}\log^2\left(\frac{1}{2}\right) \,\,$$ It follows by dividing by 2.
Dilogarithm functional equation proof
$$\mathrm{Li}_2(z) + \mathrm{Li}_{2}(1z) = \frac{\pi^2}{6}\log(z) \log(1z) \,\,\,\, ,\,0<z<1$$ $$\textit{proof}$$ Start by the following $$\mathrm{Li}_2\left(z\right) = \int^{z}_0 \frac{\log(1t)}{t} \, dt $$ Now integrate by parts to obtain $$\mathrm{Li}_2\left(z\right)= \int^z_0 \frac{\log(t)}{1t} \, dt \log(z) \log(1z) $$ By the change of variable \(t=1x … Continue reading
Posted in Dilogarithm, Polylogarithm
Tagged difference, dilogarithm, formula, functional, proof
1 Comment
Relation between Zeta and Dirichlet eta functions proof
$$\eta(s) = \left( 12^{1s} \right) \zeta(s) $$ $$\textit{proof}$$ We will start by the RHS $$\left( 12^{1s} \right) \zeta(s) = \zeta(s) – 2^{1s} \zeta(s)$$ Which can be written as sums of series $$\sum_{n=1}^\infty \frac{1}{n^s} – \frac{1}{2^{s1}}\sum_{n=1}^\infty \frac{1}{n^s}$$ $$\sum_{n=1}^\infty \frac{1}{n^s} – 2\sum_{n=1}^\infty … Continue reading