TOPICS
Search

Ramanujan's Master Theorem

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook

Suppose that in some neighborhood of x=0,

F(x)=sum_(k=0)^infty(phi(k)(-x)^k)/(k!)
(1)

for some function (say analytic or integrable) phi(k). Then

int_0^inftyx^(n-1)F(x)dx=Gamma(n)phi(-n).
(2)

These functions form a forward/inverse transform pair. For example, taking phi(k)=1 for all k gives

F(x)=sum_(k=0)^infty((-x)^k)/(k!)=e^(-x),
(3)

and

int_0^inftyx^(n-1)e^(-x)dx=Gamma(n),
(4)

which is simply the usual integral formula for the gamma function.

Ramanujan used this theorem to generate amazing identities by substituting particular values for phi(n).

See also

Glasser's Master Theorem, Ramanujan's Interpolation Formula

Portions of this entry contributed by Jonathan Sondow (author's link)

Explore with Wolfram|Alpha

References

Berndt, B. C. Ramanujan's Notebooks: Part I. New York: Springer-Verlag, p. 298, 1985.Edwards, H. M. "Ramanujan's Formula." §10.10 in Riemann's Zeta Function. New York: Dover, pp. 218-225, 2001.

Referenced on Wolfram|Alpha

Ramanujan's Master Theorem

Cite this as:

Sondow, Jonathan and Weisstein, Eric W. "Ramanujan's Master Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RamanujansMasterTheorem.html

Subject classifications