A problem with expansion

[krish]

[question]
What is the meaning of expansion of a function in calculus.What does it mean saying expansion of a function about a point.
For example i came across Taylor's expansion of a function f(x) about the point x=a which is given by
f(x)=f(a)+(x-a)f'(a)/1! + (x-a)^2*f"(a)/2!and so on ....
the formula looks simple and can be understood.But the honest question is ,what is expansion of a "function"?

[Difficulty]
the most provoking and confusing part is the word "expansion"
when it comes to expansion this is what one would think
i,e something like (x+y)^2(the whole square) ,(x+y+z)^3 and etc
how should i understand expansion in calculus and the one mentioned above ((x+y)^2(the whole square) ,(x+y+z)^3)
how is it different from the one mentioned above [i,e (x+y)^2(the whole square) ,(x+y+z)^3]

[Thoughts]
I am sorry , i understand expansion to be that of a binomial
expansion ,or in general a polynomial . i,e (x+y)^2=x^2+2xy+y^2.What is expansion of f(x) is still a question
Thank you
>Krish

[krish]

can i expect an answer for this post ......
i want to know as to what expansion is in calculus
i,e if u consider (x+y)^2=x^2+2xy+y^2... equation1
how do i obtain something on the RHS of 1.
yes by expanding it twice(i,e the power 2 in (x+y) )gives me the RHS

similarly what in f(x) gives me an infinite series.... what mathematics is used to obtain a series and how can u say that a function can be expressed as a series of different functions.
how do u get a series of functions without even letting one know as to what is being done ,in order to obtain a series

i have justified it with respect to the above example
i have justified that i can obtain x^2+2xy+y^2 only when i have multipied (x+y)twice or in other words "expand" it twice

[krish]

i hope my above post is genuine ....
Thank you
Krish

[krish]

i guess this forum doesn entertain questions from calculus ....

all physicists r good mathematicians but the converse need not be true .......
so i am posting a question in mathematics in this forum ...
so physicists .......... help me ........
pl dont delete this topic .........moderators

[deleted-71417]

Hi Krish,

I am neither a physicist nor a mathemetician, but I will take a crack at providing an answer. My dictionary defines the the mathematical useage of expansion to mean “the development at length of an expression indicated in a contracted form.” As I have seen it used it is applied primarily to series that are usualy written as summations of infinite numbers of terms, which of course you cannot evaluate. The individual terms of the series may be functions of one or more variables. Often however you can mathematically prove that these series converge quickly, to very close to the correct answer, and can prove a maximum upper bound of the maximum difference possible between the absolutely correct answer and the answer gotten by evaluating just the first few terms of the series, at least under well defined conditions. Therefore one expands the general formula to a truncated explicit sum of series elements so you can compute “good enough” answers to the value of the function represented by the series. I checked several college level calculus textbooks looking for a definition of expansion, but none listed the word in their index, so I am afraid I cannot provide a better answer. To go beyond this answer I suggest consulting a good calculus textbook, of which there are many.

Please note this forum is primarily to help students with K-12 science fair projects. Is your question related to this? At first glance it looks more like a question related to a class. If not, it would help us if you could post more information on what your project is trying to accomplish , what your grade level is, and what math/science background you have to help us tailor our answers to what you can understand.

Best wishes,

Barrett Tomlinson

[deleted-71447]

Krish,
These forums do answer questions related to calculus, but our forums are also starting to get busy this time of year, and priority goes to helping high school students with their science fair projects. Apologies if I have incorrectly assumed that your question is not related to a science fair project.
Regards,
Chris

[krish]

yes sir i can understand everything that is given but my question is how do u obtain a series
or how can i expect a function to be a sum of an infinite series .....
what mathematics is used to show that a function can be expressed as a series of diff functions
what is the idea behind
once again
i give the same example
(x+y)^2=x^2+2xy+y^2... even this is also a series also an expansion .... the mathematics or the idea that is involved here is multipication of (x+y)[twice]
What is done to f(x)in order to obtain a series is my question
Thank you sirs for ur reply
what ever replies that are seen so far is the after effect i want the starting one
The big question is how???????
once i can convince u that i can express a function in terms of a series then what ever is said so far becomes powerful defintely agreeable
But the question is how i can express a function in terms of a series .What mathematics i should adopt .
I am doing a project on power amplifiers .. i came across something called as harmonics ...... then Fourier series ... hence it starts from there .Please see my above patiently ... I am sure u will come to know as to what i am looking for
Thank you

[deleted-71417]

Hi,

The answer to your question on the foundations of the use of series in calculus is beyond my ability to answer in a short post on this site. It is a major topic in a second year college calculus course. The best I can do is to refer you to some excellent books that treat it somewhat extensively.

“Mathematics of Physics and Modern Engineering” by I.S. Sokolnikoff and R.M.Redheffer, published by McGraw-Hill Book Company of New York in 1958, Library of Congress Card Number 57-12914 is still widely available and deals with your questions in chapter 2. I have seen used copies advertised for $11 to $55, or check your local college library. This book tends to be light on formal proofs and much stronger in practical application of mathematical ideas for practicing scientists and engineers.

“Calculus Volume 1” by Tom M. Apostol, published by Blaisdell Publishing Company, New York, in 1961, Library of Congress Catalog Card Number 62-11601, is more a mathematicians view of the subject. It deals with your questions in Chapter 9, the end of the book.

These are the texts I studied from in the 1960s. I am sure there are more recent texts that cover the topics as well that are more widely available.

A quick web survey turned up this tutorial which might help:
http://tutorial.math.lamar.edu/Classes/ ... Intro.aspx

Unfortunately you may need to make a rather thorough study of calculus before any of these resources begin to make sense.

I wish you the best of luck. You have not embarked on an easy quest.

Barrett Tomlinson

[deleted-2131]

krish,

I agree with Barrett Tomlinson. The questions that you are asking are fundamental mathematical questions that deal with concepts that are very important, but rather complex. (I spent two semesters of college classes studying infinite series and representation of functions as series.) It also seems that you have ended up here in a sort of backwards way - you started with the applications and finite (?) Fourier series and are now working to a simpler level instead of starting with basics and working towards the applications.

My advice to you is two fold. First, realize that you are going to have to spend some serious time to understand the concepts needed to address the questions you are asking. Second, realize that the time you put in will pay off hugely! The resources that Barrett provided are good ones (the Apostol book is still a classic), but you might find it easier to find some newer ones. I particularly like Larson, Hostetler, and Edwards "Calculus" (ISBN 9780618606252 for the most recent edition). James Stewarts' calculus text is also popular (ISBN 978-0495011699) but I really do prefer the Larson text's exposition.

Try asking your math teacher for a calculus text. Chances are she or he has one lying around somewhere and might even be willing to help you understand things better. Don't be afraid to utilize this great resource!