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Hyperasymptotics

One case where an "expansion beyond all orders" may be needed is investigating the asymptotic behavior of the difference of two functions with coinciding dominant series.

We are interested in the asymptotic behavior of  for large positive :

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does not succeed:

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 (1)

The reason is that the dominant terms containing the factor  in the two hypergeometric functions cancel exactly, and we have to look for the subdominant terms.

The order of the leading terms can be found from :

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 (2)
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 (3)

As expected, one of the solutions (the third one for positive ) contains the  factor, the leading term being of the order .

Another, subdominant, solution is algebraic and, in fact, is a series containing only one term, as  is an exact solution. It will turn out that the algebraic part in  also cancels out.

Thus we have to look for the subsubdominant terms, which contain decaying exponentials. We will accomplish this by applying the steepest descent method to the integral representations of  and .

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 (4)
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 (5)
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 (6)

and are the integrals of  and of  over the same path, which is a loop encircling the poles of and of . Now a standard technique is to extend the integration contour far to the left, while still keeping both endpoints at . Then the arguments of the gamma functions can be made large everywhere on the integration path, and the gamma functions can be replaced by their asymptotic approximations.

When moving the contour, we have to take into account the pole of the integrand at . The other poles of  will be cancelled by the zeros of , which is why the algebraic part of the expansion will contain the single term of the order .

This is the negative of the third term in :

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 (7)

Expanding the gamma functions produces terms containing  and

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 (8)

As we shall see, those terms have saddle points  located in the left half-plane and contribute exponentially small factors . The terms for which the saddle point would be located at  have cancelled out, thus cancelling the exponentially large contributions. Another possible way to achieve the same result was to write  as a single Meijer G-function .

We write the first term above in the form :

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 (9)

For this to become zero, we need , and thus . We can visualize the paths where the imaginary part of  stays constant. The path of the steepest descent is the one that goes through the saddle point in the direction ; the blue color indicates smaller values of the real part of :

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The real part of  has a maximum along this path at .

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 (10)

Now we can compute the lead asymptotic term contributed by the saddle point :

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 (11)

We repeat the same procedure for the second term of the integrand.

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 (12)
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The direction should be chosen as  to be consistent with the direction of the integration contour, which goes from the lower to the upper half-plane.

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 (13)

Combining the two results yields the leading term of . The next terms can be obtained by expanding  and  to higher orders.

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 (14)
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