The Mellin Transform in a Nutshell

The Mellin transform is an integral transform closely related to the Fourier and Laplace transforms. Where the Fourier transform analyzes oscillations over linear shifts, the Mellin transform explores how a function behaves under scaling. For a positive variable $x$ and complex parameter $s$, it is defined by

$M\left(s\right)={\backslash int}_0{\mathrm{\backslash infty}}_{}^{}{x}^{s}f\left(x\right)\mathrm{dx}$

The transform converts multiplication of $x$ by a factor into shifts of $s$, making it useful for analyzing scale-invariant phenomena. Its applications range from number theory and asymptotic analysis to image processing and complex networks.

Numerical Evaluation

This calculator evaluates the Mellin transform for real values of $s$ using a simple change of variables $x=e^t$. The integral becomes

${\backslash int}_{-\mathrm{\backslash infty}}{\mathrm{\backslash infty}}_{}^{}{e}^{st}f\left(e^t\right)\mathrm{dt}$

We integrate numerically over a finite interval centered at zero, assuming the function decays rapidly enough. Because this is an approximation, accuracy improves for well-behaved functions and moderate values of $s$. If your integrand grows quickly or oscillates, specialized techniques may be required.

Exploring Applications

The Mellin transform plays a pivotal role in analytic number theory. For example, the Riemann zeta function can be expressed as a Mellin transform of a theta function. In image analysis, the transform helps detect scale-invariant patterns because dilations in the spatial domain correspond to translations in the Mellin domain. Researchers also use it to derive asymptotic expansions of integrals by shifting contours in the complex plane.

Because the Mellin transform reveals how functions scale, it serves as a bridge between additive and multiplicative structures. In probability theory, it is linked to the moments of positive random variables. In control theory and signal processing, analyzing transfer functions with Mellin methods uncovers how systems respond to geometric changes in frequency or size.

Working with the Calculator

Enter a function of $x$ using JavaScript syntax—such as x^2 * exp(-x)—and a real value for $s$. The script discretizes $t$ from $-10$ to $10$ with a modest step size. For each sample it evaluates $f$, multiplies by $e^{st}$, and sums the results. The final integral is scaled by the step size. While simple, this approach demonstrates the core idea and lets you experiment with different functions.