Ramanujan’s master theorem

The Endeavour 2024-11-14

Ramanujan's passport photo from 1913

A few weeks ago I wrote about the Mellin transform. Mitchell Wheat left comment saying the transform seems reminiscent of Ramanujan’s master theorem, which motivated this post.

Suppose you have a function f that is nice enough to have a power series.

$f(z) = \sum_{k=0}^\infty a_k z^k$

Now focus on the coefficients a_n as a function of k. We’ll introduce a function φ that yields the coefficients, with a twist.

$f(z) = \sum_{k=0}^\infty \frac{\varphi(k)}{k!} (-z)^k$

and so φ(k) = (−1)^k k! a_k. Another way to look at it is that f is the exponential generating function of (−1)^k φ(k).

Then Ramanujan’s master theorem gives a formula for the Mellin transform of f:

$\int_0^\infty z^{s-1} f(z) \, dz = \Gamma(s) \varphi(-s)$

This equation was the basis of many of Ramanujan’s theorems.

Master theorem of algorithm analysis
Ramanujan’s approximation for the perimeter of an ellipse
Transforms and convolutions

The post Ramanujan’s master theorem first appeared on John D. Cook.

Ramanujan’s master theorem

The Endeavour 2024-11-14

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