Best of Today's Calculation

Here are chosen ones from my 50+ε calculations from the Today's Calculation archive, which are polished and supplemented.

Any feedback is greatly welcomed.

Todays_Calculation.pdf

Table of Contents

1. Preliminary
1. Some materials on analysis
2. Mathematical Constants
3. Riemann Zeta function
4. Gamma function and Polygamma functions
5. Beta function
6. Polylogarithms and related functions
2. Today’s Calculation
1. Derivation of $\zeta(2n)$
2. An integral related to the Euler-Mascheroni constant
3. Legendre chi function
4. Fourier transform of the extended binomial coefficient
5. Alternating Euler sum involving binomial $\binom{2n}{n}$
6. An Integral with logarithmic behavior
7. A tough Euler integral
1. Reduction to Euler sum
2. Evaluation of $A^{(s)}_{m}$
3. Principal Value of the Polylogarithm
4. Conclusion
8. A logarithmic singularity renormalization
9. A Product related to Dedekind eta function
10. An integral related to the Dirichlet beta function
11. An Limit of a Probabilistic Sum
12. An Probabilistic Limit