Alternative to Arfken Regarding some Chapters

I don't understand what is confusing about the examples you gave. The first is just multiplying out the integrand. In the second, the factor of sqrt(2pi) is just convention. Surely any reading of Fourier transforms would have mentioned that. Are you sure you've actually been paying attention to what you're reading?

Wikipedia is actually a pretty good source for mathematics; although when reading an article you'll have to take careful note of the conventions used, because they may be different than what you are used to.

Another good reference for the confusing bits of Arfken is, believe it or not, previous editions of Arfken. In the latest edition, they have tried to reduce the size of the book, and the editing was done poorly. Whole pages of auxiliary explanations and examples were simply cut out, mostly in the more esoteric parts of the complex analysis sections (such as asymptotic series, product expansions, pole expansions, and the saddle point method). In some cases, this has resulted in gaps in the line of reasoning which make the book difficult to follow.

When reading a book like Arfken, make sure you work along with it rather than just read. Things will be much clearer that way.

 

The "why" is something you have to learn by experience. Most books don't bother explaining why they choose to do one manipulation rather than some other manipulation. You have to practice and get a feel for what manipulations lead in the direction you want, and what manipulations don't. This is ultimately the "why" of it. Good foresight as to how to manipulate expressions is part of what separates good mathematicians from others.

Because now all the imaginary numbers are outside the integrals, and we can apply what we know from the calculus of functions of real numbers.

In your latest example, have you never seen a geometric series before?

[tex]\sum_{k=0}^\infty r^k = \frac{1}{1-r}[/tex]

in general (whenever the series converges). If you don't understand that, try putting r=1/2, and you should be able to see why it works.

 

The geometric series is something simple and easy that you should have learned by heart a long time ago. I'm not sure how you got to complex analysis without ever becoming familiar with it. It's not enough to say, "Oh, I can just look that up in a book somewhere".

If you truly want to find shortcuts and become fast at manipulating expressions, then you have to practice and learn this kind of thing. Geometric series, binomial series, Taylor/Maclaurin series for e^x, ln(1+x), sin(x) and cos(x), trig identities, hyperbolic trig identities, law of cosines, etc. All that stuff you used to think you could just look up somewhere...if you are unable (or unwilling) to keep them in the active part of your mind, then you will miss some extraordinary shortcuts!

Arfken is trying to show you exactly one of those extraordinary shortcuts: How to calculate Laurent/Taylor series quickly and efficiently for any rational function. The method is to split the function into partial fractions, and then make each term look like a geometric series. It's much faster than explicitly calculating derivatives or integrals to get the series coefficients.

You seem to be more interested in asking "What's the trick I should use here?" instead of trying to get a better understanding. But the "trick" is to use the understanding you should have gotten from previous courses! Nothing in mathematics happens in isolation.

 

I already gave you an answer to that question.