## Asymptotic expansions of integrals

I consider several examples of deriving asymptotic expansions of real and complex integrals. Such expansions are important in physics and widely used. But the derivation process often involves multiple stages, tricky changes of variable and tedious, error-prone calculations. I struggled with this subject as a student. In retrospect, I think my main problem was not being able to mentally visualize important aspects of the derivation. In the following notebooks, I take full advantage of the visualization capabilities of Mathematica to illustrate geometrical features such as steepest descent paths, saddle points, and contour deformations. Of course, having access to symbolic mathematics capabilities and the ability to cross-check intermediate analytical results against numerics also facilitates the derivations.

The following examples are taken from one of my favorite books – *Advanced Mathematical Methods for Scientists and Engineers*, Carl M. Bender and Steven A. Orszag. I also highly recommend the magnificent series of lecture videos by Carl Bender.

**17-15 BenderOrszag Ex1p241 03-31-16**

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**17-16 BenderOrszag Ex2p253 09-14-15**

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**17-17 BenderOrszag Ex2p283 04-11-16**

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**17-18 BenderOrszag Ex4-5-6-7 p289 04-18-16**

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**17-19 BenderOrszag Ex8 p291 04-21-16**

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**17-20 BenderOrszag Ex09 p294 04-30-16**

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**17-21 BenderOrszag Ex10 p296 04-30-16**

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**17-22 BenderOrszag Ex11 p298 07-18-16**

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**17-23 BenderOrszag Ex12 p299 07-20-16**

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**17-38 Asymptotic analysis of Airy Eqn 07-23-16**

I work through the integral representation of the Airy function, and the generation of asymptotic expansions using the method of steepest descent. Contours of integration have to be deformed through saddle points. Various visualizations of the analytical procedure are generated. Various manipulations of complex expressions and changes of variable are invoked.

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