Differentiation Under the Integral Sign

TeeEm

19

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(edited 7 years ago)

Reply 1

mik1a

10

This is a very neat trick, bumping to increase awareness of it.

PS. is there any way you can enlarge your scanned workings? Just looking at it, someone printing these sheets will probably find the outputted text rather small.

Reply 2

Kvothe the Arcane

20

Original post by mik1a

This is a very neat trick, bumping to increase awareness of it.

PS. is there any way you can enlarge your scanned workings? Just looking at it, someone printing these sheets will probably find the outputted text rather small.

I think he has the workings purposefully small. That being said, the resolution is decent so zooming doesn't destroy the quality.

Reply 3

DFranklin

18

TeeEm

..

Have you seen the way you can evaluate the gaussian integral using this kind of approach? (See half way down page 4 of http://www.math.uconn.edu/~kconrad/blurbs/analysis/diffunderint.pdf)

It's a bit more complicated but I think it's worth exposing people to.

Reply 4

TeeEm

OP

19

Original post by DFranklin

Have you seen the way you can evaluate the gaussian integral using this kind of approach? (See half way down page 4 of http://www.math.uconn.edu/~kconrad/blurbs/analysis/diffunderint.pdf)

It's a bit more complicated but I think it's worth exposing people to.

It is a good one but not the best introduction.

I already have it but I decided to stick to 10 examples this time, otherwise I might as well move my site into tsr.

Original post by TeeEm

It is a good one but not the best introduction.

It's not the best introduction, but at the same time I think the examples that show "here's how to do an important integral that you could never do in a conventional manner" are the ones that stick in people's minds. (Certainly the sin t / t and exp(-x^2) integrals are the ones I remember from times past).

[But I should add that I did very much appreciate this set of examples - TSR's "please rep some other member" policy has stopped me giving the rep you deserve!]

I already have it but I decided to stick to 10 examples this time, otherwise I might as well move my site into tsr.

Reply 6

shamika

16

Original post by TeeEm

It is a good one but not the best introduction.

I already have it but I decided to stick to 10 examples this time, otherwise I might as well move my site into tsr.

I think I prefer the first few examples in the Conrad paper to yours. For example, it isn't entirely obvious to someone without any knowledge of differentiating under the integral that you should introduce another parameter (as in your first example).

Having said that, I do think these examples are massively helpful for help / revision of methods type courses at uni so I don't want to discourage you from making and distributing them!

It helps that Conrad actually has some explanation, whereas you obviously don't because you're not writing a paper / textbook. I thought he wrote very clearly - for example

This is called differentiation under the integral sign. If you are used to thinking mostly about functions with one variable, not two, keep in mind that (1.2) involves integrals and derivatives with respect to separate variables: integration with respect to x and differentiation with respect to t.

will help a lot of people get over their initial confusion pretty quickly in my opinion.

(edited 8 years ago)

Reply 7

TeeEm

OP

19

Original post by shamika

I think I prefer the first few examples in the Conrad paper to yours.

You are entitled to your opinion.
You are further entitled to publicise your opinion.

Original post by shamika

I don't want to discourage you from making and distributing them!

I do not think that in my ripe age now, mathematics education to postgrad level, 30 years experience in teaching and lecturing mathematics I would be discouraged from the opinion of some forum users. Besides these resources are not produced for TSR but for my own purposes.

all the best

Reply 8

shamika

16

Original post by TeeEm

You are entitled to your opinion.
You are further entitled to publicise your opinion.

I do not think that in my ripe age now, mathematics education to postgrad level, 30 years experience in teaching and lecturing mathematics I would be discouraged from the opinion of some forum users. Besides these resources are not produced for TSR but for my own purposes.

all the best

I'd be interested in what you think is the best introduction to differentiation under the integral sign, as I thought Conrad's is pretty good.

Reply 9

TeeEm

OP

19

Original post by shamika

I'd be interested in what you think is the best introduction to differentiation under the integral sign, as I thought Conrad's is pretty good.

It is honestly a tough book/booklet to start with but it has some amazing techniques.
I can think of other classic books such as "Mathematical methods for science students" by Stephenson to introduce the topic.
Opinions can vary.

PS
My work/resources are not books to introduce topics but worked out problems for students that have some introductory/basic grasp of the topic in question.

I had a message from a keen sixth former which commented that he could not follow Fourier series from my work! Well if he could I would be very surprised.

Reply 10

TeeEm

OP

19

for those you want a few more,
My site now contains
36 questions on differentiation under the integral sign
and
4 questions on integration under the integral sign

(still have a few to add but if you come across some novelty, please let me know)

Reply 11

SCAR H

2

Original post by TeeEm

for those you want a few more,
My site now contains
36 questions on differentiation under the integral sign
and
4 questions on integration under the integral sign

(still have a few to add but if you come across some novelty, please let me know)