ps06sol

ps06sol - 18.03 Problem Set 6 Fall 2009 Solutions In the...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 18.03 Problem Set 6 Fall 2009 Solutions In the entire problem set, f will be a function defined for all real numbers t , taking usually real values (but complex values would work fine). I’ll always assume that f is “piecewise continuous” in the sense explained in section 8.1 of the text; this will ensure that the integrals I write down make sense. Finally, f will always be assumed to be periodic of period 2 π . There are three kinds of “Fourier coefficients” a n = 1 π Z π- π f ( t )cos( nt ) dt ( n = 0 , 1 , 2 ,... ) . b n = 1 π Z π- π f ( t )sin( nt ) dt ( n = 1 , 2 , 3 ,... ) . c n = 1 2 π Z π- π f ( t ) e- int dt ( n = 0 , ± 1 , ± 2 ... ) . (If it isn’t clear what function f is meant, I’ll write a n ( f ) and so on.) This leads to three kinds of Fourier series: a / 2 + ∞ X n =1 a n cos( nt ) (Cosine series of f ) ∞ X n =1 b n sin( nt ) (Sine series of f ) c + ∞ X n =1 ( c n e int + c- n e- int ) (Complex series of f ) 1. (5 points) a) Write a formula expressing c in terms of a . We have c = 1 2 π Z π- π f ( t ) dt and a = 1 π Z π- π f ( t ) dt, so c = a / 2 . b) For n = 1 , 2 , 3 ,... , write a formula expressing c n and c- n in terms of a n and b n , and explain why your formula is correct. c) For n = 1 , 2 , 3 ,... , write a formula expressing a n and b n in terms of c n and c- n , and explain why your formula is correct. Note that e- int = cos( nt )- i sin( nt ) and e- i (- n ) t = cos( nt ) + i sin( nt ) . Thus, for n = 1 , 2 , 3 ,... , we have c n = ( a n- ib n ) / 2 and c- n = ( a n + ib n ) / 2 . Adding and subtracting these equations, we get e- int + e- i (- n ) t = 2cos( nt ) and e- int- e- i (- n ) t =- 2 i sin( nt ) ....
View Full Document

Page1 / 5

ps06sol - 18.03 Problem Set 6 Fall 2009 Solutions In the...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online