By Jerrold Marsden, Alan Weinstein

This can be the second one publication of a three-volume paintings referred to as "Calculus" by means of Jerrold Marsden and Alan Weinstein. This publication is the outgrowth of the authors' event instructing calculus at Berkeley. It covers innovations and purposes of integration, limitless sequence, and differential equations. in the course of the booklet, the authors encourage the learn of calculus utilizing its purposes. Many solved difficulties are incorporated, and vast routines are given on the finish of every part. additionally, a separate scholar advisor has been ready.

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**Extra resources for Calculus 2**

**Example text**

7]. 5 General infinite sums of functions As we have seen, a power series consists of infinite sums of terms anx n , n E N. Since not all functions have a representation of this type, it is natural also to consider infinite sums of other types of "simple" functions. More generally, we will consider a family of functions 10,11, h, ... with the same domain of definition I, and attempt to define the function = lo{x) + 11 (x) + h{x) + ... + In(x) + ... = L: In{x). 23) n=O Here it is of course important to know for which values of x the expression for I{x) makes sense: this is the case exactly for the x E I for which the series 01 numbers 2:~=0 In{x) is convergent.

7 Signal transmission 45 then all information about the function f is stored in the coefficients {an}~o. 27) to find out which signal we are dealing with. Let us explain how this can be used in signal transmission. Assume for example that a sender S wishes to send a picture to a receiver n. For simplification, we assume that the picture is given as a graph of a function f, and that f can be represented as a power series. Then the transmission can be done in the following way: • S finds the coefficients ao, al, ...

For simplification, we assume that the picture is given as a graph of a function f, and that f can be represented as a power series. Then the transmission can be done in the following way: • S finds the coefficients ao, al, ... such that f(x) = 2:~=o anxn; • S sends the coefficients ao, al , ... ; • n receives the coefficients ao, al,"'; • n reconstructs the signal by multiplying the coefficients an by xn and forming the infinite series f(x) = 2:~=o anxn. In practice there are a few more steps not described here.