Ramanujan’s master theorem

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.

Now focus on the coefficients a_k as a function of k. We'll introduce a function that yields the coefficients, with a twist.

and so (k) = (-1)^k k! a_k. Another way to look at it is thatf is the exponential generating function of (-1)^k (k).

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

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

Related posts

• 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.