Fourier Series question - finding coefficients

**sanka23** · July 27th 2011, 03:44 AM

Find the Fourier coefficients of the following equation and write the function as infinite series.

V = 20 + 10 sin t

**mr fantastic** · July 27th 2011, 05:01 AM

Originally Posted by sanka23

Find the Fourier coefficients of the following equation and write the function as infinite series.

V = 20 + 10 sin t

What have you tried? Where are you stuck?

**sanka23** · July 27th 2011, 05:12 AM

so far im hovering around these values;

-5i n = 1
5i n = -1
20 n = 0

but I dont really understand what it is asking me to do when it says ' write the function as infinite series'. Am I just totally off track?

**HallsofIvy** · July 27th 2011, 06:24 AM

Whether you are "completely off base" depends on exactly what you think a "Fourier series" is! A Fourier series is an infinite series of the form
$\sum_{n= 0}^\infty a_n sin(nx)+ b_n cos(nx)$

However, it is quite possible for most of those coefficients, $a_n$ and $b_n$ , to be 0! In this case, you don't have to do any "calculation" at all- just compare
$\sum_{n= 0}^\infty a_n sin(nx)+ b_n cos(nx)$
and 20+ 10 sin(t).

**MSUMathStdnt** · August 13th 2011, 02:51 PM

Originally Posted by HallsofIvy

Whether you are "completely off base" depends on exactly what you think a "Fourier series" is! A Fourier series is an infinite series of the form
$\sum_{n= 0}^\infty a_n sin(nx)+ b_n cos(nx)$

However, it is quite possible for most of those coefficients, $a_n$ and $b_n$ , to be 0! In this case, you don't have to do any "calculation" at all- just compare
$\sum_{n= 0}^\infty a_n sin(nx)+ b_n cos(nx)$
and 20+ 10 sin(t).

Hi. I'm trying to learn Fourier Series (and the transform, too). This thread touches on a couple of questions I have, may I post them here?

The definition I have for the Fourier Series, $g(t)$ (with $P$ as the fundamental period), is:
$g(t)=\sum_{n=-\infty}^{\infty} c_n \cdot e^{i\frac{2 \pi n t}{P}}$

where $c_n$ is (using $v(t)$ as the original equation from the OP):
$c_n=\frac1T \int_0^T v(t) \cdot e^{-i\frac{2 \pi n t}{P}}dt$

I can use Euler's identity to get your definition, but you are starting your series at $0$ , the definition I have starts at $-\infty$ . Why the indexing difference?

Using your definition, shouldn't $a_n=b_n$ for all $n$ ? If your definition comes from my definition and Euler's identity, the coefficients would be the same.

Last, I tried (for practice) the problem from the OP, using $P=\pi$ as the fundamental period). I tried comparing $v(t)=20+10sin(t)$ to your series definition. I get:
$a_1=10$
$a_n=0\; \forall n \ne 1$
$b_n=0\; \forall n$ .

Is that right? If so, what is the final answer (the infinite Fourier Series)? What about the $20$ in original equation?

I also tried the definition I have for $c_n$ . But those integrations return very complicated answers (I used Maple to calculate them), and nothing like what I have in the last paragraph. Why doesn't that integration turn up the same results?

Thanks, in advance, for anyone who replies.

**Mondreus** · August 14th 2011, 10:17 AM

HallsOfIvy posted the definition of a fourier series on real form. You have the complex form.

Complex Form of Fourier Series

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