New Blog: Advanced Integration: Math, Integration, Special Functions by Zaid Alyafeai

Zaid Alyafeai has started a blog: Advanced Integration: Math, Integration, Special Functions. He is the author of Advanced Integration Techniques, which I highly recommend, and he is active on Mathematics Stack Exchange.

Advanced Integration Techniques by Zaid Alyafeai

Zaid Alyafeai has published a new version of his free e-book Advanced Integration Techniques. He has added about 50 pages of new material. From the table of contents, it appears that he had added the following chapters:

  • Sine Integral function
  • Cosine Integral function
  • Integrals involving Cosine and Sine Integrals
  • Logarithm Integral function
  • Clausen functions
  • Clausen Integral function
  • Barnes G function

There are many interesting results collected here that are otherwise difficult to find as they are scattered in many different references. As such, the book is not only an excellent source of integration methods, it is also one for the practical use of many special functions. I enthusiastically recommend Alyafeai’s book to readers of this blog.

Derivation of an Integral Expression of the Euler-Mascheroni Constant – Part 2

I derived the expression
\begin{equation}
\int\limits_{0}^{\infty} \mathrm{e}^{-x} \mathrm{ln}(x) \mathrm{d} x = -\gamma
\label{eq:1608161}
\tag{1}
\end{equation}
for the Euler-Mascheroni constant here. However, there is a far easier method that was fully derived in Advanced Integration Techniques by Zaid Alyafeai. I recommend this book to readers of this blog. It is free and contains many useful and interesting results.

We start with
\begin{equation}
\int\limits_{0}^{\infty} \mathrm{e}^{-x} x^t \mathrm{d} x = \Gamma(t+1)
\label{eq:1608162}
\tag{2}
\end{equation}
Differentiate with respect to \(t\)
\begin{equation}
\int\limits_{0}^{\infty} \mathrm{e}^{-x} x^t \mathrm{ln}(x) \mathrm{d} x = \frac{d\Gamma(t+1)}{dt} = \Gamma(t+1) \psi^{(0)}(t+1)
\label{eq:1608163}
\tag{3}
\end{equation}

Taking the limit of equation \eqref{eq:1608163}, \(t \to 0\) yields
\begin{equation}
\int\limits_{0}^{\infty} \mathrm{e}^{-x} \mathrm{ln}(x) \mathrm{d} x = \Gamma(1) \psi^{(0)}(1) = -\gamma
\label{eq:1608164}
\tag{4}
\end{equation}