Chapter 7: Q6MP (page 387)

Let $f (t) = e^{i ω t}$ on $(- π, π)$ . Expand $f (t)$ in a complex exponential Fourier series of period $2 π$ . (Assume $w \neq$ integer.)

Short Answer

Expert verified

The expanded function $f (t)$ in a complex exponential Fourier series of period $2 π$ is $f (t) = \sum_{n = - \infty}^{\infty} \frac{\sin (π (ω - n))}{π (ω - n)} e^{int}$ .

Step by step solution

Given information

The given function is $f (t) = e^{i w t}$ . The function $f (t)$ is to be expanded in a complex exponential Fourier series of period $2 π$ .

Write complex series formulas

The following are the formulas for the complex Fourier series,

$\begin{matrix} f (t) = \sum_{- \infty}^{\infty} c_{n} e^{i n π t / l} \\ c_{n} = \frac{1}{2 l} \int_{- l}^{l} f (t) e^{- i n π t / l} d t \end{matrix}$

Here the period of $f (t)$ is $2 l$ and the frequencies of the terms in the series are $n / 2 l$ .

Find the complex exponential Fourier series

First evaluate complex Fourier series coefficients of the series.

$\begin{matrix} c_{n} = \frac{1}{2 π} \int_{- π}^{π} e^{i ω t} e^{- int} d t \\ = \frac{1}{2 π} \int_{- π}^{π} e^{i t (ω - n)} d t \\ = {\frac{1}{2 π} \frac{1}{i (ω - n)} e^{i t (ω - n)} |}_{- π}^{π} \\ = \frac{1}{2 π} \frac{2 i \sin (π (ω - n))}{i (ω - n)} \end{matrix}$

Further solve to get $c_{n}$ .

$c_{n} = \frac{\sin (π (ω - n))}{π (ω - n)}$

Now, evaluate the function $f (t)$ .

$f (t) = \sum_{n = - \infty}^{\infty} \frac{\sin (π (ω - n))}{π (ω - n)} e^{int}$

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Let $f (t) = e^{i ω t}$ on $(- π, π)$ . Expand $f (t)$ in a complex exponential Fourier series of period $2 π$ . (Assume $w \neq$ integer.)

Short Answer

Step by step solution

Given information

Write complex series formulas

Find the complex exponential Fourier series

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Let f(t)=eiωton(−π,π). Expandf(t)in a complex exponential Fourier series of period 2π. (Assume w≠integer.)

Short Answer

Step by step solution

Given information

Write complex series formulas

Find the complex exponential Fourier series

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Mechanics and Materials

Quantum Physics

Atoms and Radioactivity

Radiation

Magnetism

Force

Study anywhere. Anytime. Across all devices.

Let $f (t) = e^{i ω t}$ on $(- π, π)$ . Expand $f (t)$ in a complex exponential Fourier series of period $2 π$ . (Assume $w \neq$ integer.)