- #1

Sheridans

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## Homework Statement

Hello,

Check each function to see whether it is piecewise smooth. If it is, state the value to which its Fourier series converges at each point x in the given interval and the end points

(a.) f(x)=|x|+x, -1<x<1

(it would be very helpful to see if i did this right, as the professor I have does not do examples and that is how I learn how to approach and solve problems)

## Homework Equations

If f is piecewise smooth and is periodic with a period of 2a, then at each point x in the corresponding Fourier series to f converges and its sum is:

Fourier series= 0.5(f(x+)+f(x-)), where f(x+) is the limit from the right, and f(x-) is the limit from the left.

Criterion for piecewise smooth on interval a<x<b:

1) f is piecewise continuous (it is bounded and is continuous, except possibly for a finite number of jumps and removable discontinuities)

2)f'(x) exists except possibly at a finite number of points

3) f'(x) is piecewise continuous

## The Attempt at a Solution

After sketching the function, it is continuous on the interval, f'(x) exists and it has a finite number of discontinuities (so it is piecewise continuous) Therefore, f(x) is piecewise smooth

f(x)= 2x, 0<x<1

0, -1<x<0 (spilt it up)

f(x)=.5(f(x+)+f(x-))=2x/2=x

endpoints: at x=-1 .5(f(-1+)+f(-1-))=-2/2=-1

at x=1, .5(f(1+)+f(1-))=2/2=1

Is this right? I think i went wrong somewhere. And do i have to actually find the Fourier series (which I know how to do, just thought it was not needed/was not even specified)?

## Homework Statement

## Homework Equations

## The Attempt at a Solution

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