Let f have fourier series fx fraca02 sumn0 an cosnx bn sinnx...

Let $f$ have Fourier series $$f(x) = \frac{a_0}{2} + \sum_{n>0} a_n \cos(nx) + b_n \sin(nx).$$ Show that this can be re-written as $$f(x) = \sum_{n \in \mathbb{Z}} c_n e^{inx}$$ where $$c_n = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) e^{-inx} dx.$$ Any of the other exercises in section 33 are helpful practice.

Transcript

00:01 F of x is given by half a not plus the sum 1 to infinity an n cosine x plus some 1 to infinity b n sine x.

00:10 It is also represented as a sum negative to infinity to infinity c n, e to the inx.

00:16 Use euler's formula to find a .n and bn in terms of c and negative and c .n.

00:21 And then to find cn and c negative n in terms of a .n and b .m.

00:27 So we have that the sum negative infinity to infinity to infinity.

00:35 C n e to the i n x is the sum negative infinity to infinity c n then it becomes cosine nx plus i sign nx so that's going to be the sum okay actually let's start with the first the zero term so that's going to be c zero then it's going to be cosine of zero is one and sign of zero is zero so which is c0 plus the sum 1 to infinity cn cosine n x plus i sign nx plus i sign nx plus the sun negative infinity to negative 1 c n cosine n x plus i sign n x okay now here i want to just negate the index okay, so this is n is going from negative infinity to negative one.

01:47 Now i want n to go from one to infinity.

01:49 That means that this n becomes negative, right? so this will be c0 plus the sum one to infinity, c n, cosine nx plus i sign x.

02:05 This is going to be the sum one to infinity of c negative n, cosine negative nx plus i sign negative nx.

02:16 Okay, now cosine of a negative is equal to cosine of the positive.

02:22 So this is just cosine nx.

02:25 While sine of a negative is the negation of sign of the positive, so this is negative i sine x.

02:33 Okay, so this becomes c not plus the sum, i is one, sorry, n is one to infinity.

02:40 Okay, i've got c n cosine nx, and i've also got c negative n of cosine nx.

02:47 So that's cn plus c negative n, cosine, nx...

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