Feynman integration trick - textbooks?

[Thorstein]

"Feynman integration trick" - textbooks?

I was quite impressed by the use of the parametric differentiation e.g. in integrating sin(x)/x in https://www.physicsforums.com/showthread.php?t=80735. Can anyone recommend textbooks that cover this use as a strategy in evaluating integrals? Ideally with some elementary problems for getting some practice at this.

 

[mathwonk]

there are many references in the thread you referenced. but note this is NOT a technique for finding antiderivatives, merely for evaluating very specific definite integrals. e.g. the trick does not integrate sin(x)/x over every interval, but just from minus infinity to infinity.

 

[ice109]

yes contour integration, but it looks like differentiation under the integral sign is a trick for finding anti derivatives but actually it just looks to me like substitution

 

[Gib Z]

yes contour integration, but it looks like differentiation under the integral sign is a trick for finding anti derivatives but actually it just looks to me like substitution

Whats contour integration got to do with this :S?

It indeed is Differentiation Under the Integral Sign, which is actually used for finding definite integrals, not anti derivatives. I myself have not learned this method off a textbook but from many examples on the internet. Go on Wikipedias article for this and you get some good examples. I doubt many textbooks will have this, and if they do, no more than a few pages on this, because this technique applies to very specific cases.

 

[ice109]

i'm referencing this post: https://www.physicsforums.com/showpost.php?p=620616&postcount=5

where he doesn't mention limit for the integral of xcos(ax). is this technique the same thing as parametric differentiation to solve integrals?

like this ?

 

[nicksauce]

I learned this technique (differentiating under the integral) in Adams' calculus book, although I believe the section on it was short.
https://www.amazon.com/dp/0321270002/?tag=pfamazon01-20
If I recall correctly there are also quite a few examples in the Schaums' advanced calculus book.

 

[Gib Z]

ice109- Yea parametric differentiation is another name for it. I guess I forgot that you indeed can use it for anti derivatives sometimes. That video you posted is a good example of this techinique, though he math was just a tiny bit edgy in some places. Actually I guess he made no (crucial) errors, but he could have explained it better :(

 

[lurflurf]

I do not know otf any books that cover this in much depth.
Few books are very rigorous, probably because it is difficult be rigorous and cover all the common examples.
Many books have brief coverage and some problems/examples.

Like in other areas many books give examples similar/idenical to those in others

ie
sin(x)/x
exp(-x^2)
sin(x^2)
exp(-x^2+a/x^2)

applied books
-ie math for some subject books
books an analysis/calculus
-calculus
-advanced calculus
-real analysis
-complex analysis

in particular
-Advanced Calculus, by Frederick Shenstone Woods
The above mentioned book that Feynman read and became enamored with the technique
-Excursions in Calculus: An Interplay of the Continuous and the Discrete (Dolciani Mathematical Expositions) by Robert M. Young
has some related stuff as well as cute things from other parts of calculus

 

[Thorstein]

Thanks a lot for that all. I got hold of Advanced Calculus by Woods and am working through that, will take a look at Adams and Young also.

Schaum's had as you say a few problems, you can find it in Google books : http://books.google.co.uk/books?id=...ntcover&vq=schaum's+advanced+calculus#PPA1,M1.

Mathworld also covers this briefly (including the opposite operation - integrating under the integral sign).

 

[yasiru89]

Needham explains it very briefly in Visual Complex Analysis as an alternative to contour integration but for some specific(but very popular) problems. If you work with a generalised form the 'parametric differentiation' trick is somewhat obvious with a little thought!

 

[nanoWatt]

Man, I have been away a long time. It's been 7 years since I worked with Calculus. Now I'm just going back to a community college to take Cal I again.

Is this Feynman stuff covered in any of the Calculus classes nowadays? I don't recall if it was then. I just feel so lost in the higher math, when Calculus starts to become interesting.

 

[quantumdude]

I first learned how to parametrize an integral like that from Matthews and Walker, Methods of Theoretical Physics.

 

[Gib Z]

Man, I have been away a long time. It's been 7 years since I worked with Calculus. Now I'm just going back to a community college to take Cal I again.

Is this Feynman stuff covered in any of the Calculus classes nowadays? I don't recall if it was then. I just feel so lost in the higher math, when Calculus starts to become interesting.

I don't think you'll run into it in Calc I, but you never know =] If you do, you definitely won't be spending very long on it, as it applies to very specific forms of integrals and more general methods can solve the same integrals, though the method may be more advanced. It's like the easier case you'll run into very soon - You can find the area under a parabola by taking a summation and using a special expression for one of the terms to evaluate the area. However, we had to use this special expression that only applied there. Later you find out you can find the area under any general polynomial through a more general method that applies to different things as well. Just like this with contour integration, sort of.

 

[Mute]

Is this Feynman stuff covered in any of the Calculus classes nowadays? I don't recall if it was then. I just feel so lost in the higher math, when Calculus starts to become interesting.

It's come up in my classes. Never really as a topic in its own right, but more just as a trick to evaluate certain integrals. Of course, those "certain integrals" were always Gaussian integrals, but it was always much easier to take derivatives than to prove things by contour integration again and again and again.

Once you prove

$$\int_{-\infty}^{\infty}dx~e^{-\alpha x^2} = \sqrt{\frac{\pi}{\alpha}}$$

you can just differentiate with respect to $\alpha$ again and again and again to evaluate integrals of the form

$$\int_{-\infty}^{\infty}dx~x^{2n}e^{-\alpha x^2}$$