Borel summation - Definition - Encyclopedia II

Let be a formal power series in z. Define the Borel transform of y by . Suppose that has a nonzero radius of convergence as a function of t can be analytically continued to a function on all of the positive real line grows at most exponentially along the positive real line Then the Borel sum of y is given by the Laplace transform of . This function ...

Borel summation, Borel summation - Applications, Borel summation - Discussion

Let

be a formal power series in $z$ .

Define the Borel transform of $y$ by

Suppose that

has a nonzero radius of convergence as a function of $t$
can be analytically continued to a function on all of the positive real line
grows at most exponentially along the positive real line

Then the Borel sum of $y$ is given by the Laplace transform of . This function is guaranteed to exist by condition (3) above.

Adapted from the Wikipedia article "Definition", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki/Main_Page

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