The Reflection Identity of the Gamma Function via Complex Integration

Sun, 15 Feb 2026 23:35:59 -0600

The reflection identity of the gamma function is:

\begin{equation} \Gamma( z ) \Gamma( 1-z ) = \frac{\pi}{\sin( \pi z ) } \end{equation}

We can start from the definition of $\beta( 1- \alpha, \alpha ) $

\begin{align} \Gamma( 1- \alpha ) \Gamma( \alpha ) &= \beta ( 1- \alpha, \alpha ) = \int^\infty_0 \frac{t ^{1- \alpha-1}dt}{( 1+ t )^{1- \alpha+ \alpha} } \\ &= \int^\infty_0 \frac{t ^{-\alpha} dt}{1 + t} \\ &= \int^\infty_{-\infty} \frac{e ^{-\alpha x} e^x dx}{1+ e^x} = \int^\infty_{-\infty} \frac{e ^{( 1-\alpha )x }}{ 1 + e^{x}} = I \end{align}

General Lorentz Transformation via Group Generators

Thu, 12 Feb 2026 15:52:52 -0600

In this post we derive the functional form of a generic Lorentz transformation from its group generators. At first glance I considered this trivial, as I was unaware of the role played by their non-commutativity — which made it an interesting exercise to present. We follow Chapter 11 of Jackson (Third Edition), which makes use of the mathematical properties of Minkowski spacetime. Clarifications missing from the book are supplied with original arguments.

Blog Posts on Edgar Ivan Hinojosa

The Reflection Identity of the Gamma Function via Complex Integration

General Lorentz Transformation via Group Generators