Q6MP Let f(t)=ei蝇t聽on聽(鈭捪€,蟺). ... [FREE SOLUTION]

Chapter 7: Q6MP (page 387)

Let $f (t) = e^{i 蝇 t}$ on $(鈭� 蟺, 蟺)$ . Expand $f (t)$ in a complex exponential Fourier series of period $2 蟺$ . (Assume $w 鈮�$ integer.)

Short Answer

Expert verified

The expanded function $f (t)$ in a complex exponential Fourier series of period $2 蟺$ is $f (t) = {鈭�}_{n = 鈭� 鈭�}^{鈭�} \frac{\sin (蟺 (蝇鈭� n))}{蟺 (蝇鈭� n)} e^{int}$ .

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 $2 蟺$ .

Write complex series formulas

The following are the formulas for the complex Fourier series,

$\begin{matrix} f (t) = {鈭�}_{鈭� 鈭�}^{鈭�} c_{n} e^{i n 蟺 t / l} \\ c_{n} = \frac{1}{2 l} {鈭�}_{鈭� l}^{l} f (t) e^{鈭� i n 蟺 t / l} d t \end{matrix}$

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.

$\begin{matrix} c_{n} = \frac{1}{2 蟺} {鈭�}_{鈭� 蟺}^{蟺} e^{i 蝇 t} e^{鈭� int} d t \\ = \frac{1}{2 蟺} {鈭�}_{鈭� 蟺}^{蟺} e^{i t (蝇鈭� n)} d t \\ = {\frac{1}{2 蟺} \frac{1}{i (蝇鈭� n)} e^{i t (蝇鈭� n)} |}_{鈭� 蟺}^{蟺} \\ = \frac{1}{2 蟺} \frac{2 i \sin (蟺 (蝇鈭� n))}{i (蝇鈭� n)} \end{matrix}$

Further solve to get $c_{n}$ .

$c_{n} = \frac{\sin (蟺 (蝇鈭� n))}{蟺 (蝇鈭� n)}$

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

$f (t) = {鈭�}_{n = 鈭� 鈭�}^{鈭�} \frac{\sin (蟺 (蝇鈭� n))}{蟺 (蝇鈭� n)} e^{int}$

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91影视

Let $f (t) = e^{i 蝇 t}$ on $(鈭� 蟺, 蟺)$ . Expand $f (t)$ in a complex exponential Fourier series of period $2 蟺$ . (Assume $w 鈮�$ integer.)

Short Answer

Step by step solution

Given information

Write complex series formulas

Find the complex exponential Fourier series

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Most popular questions from this chapter

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91影视

Let f(t)=ei蝇ton(鈭�蟺,蟺). Expandf(t)in a complex exponential Fourier series of period 2蟺. (Assume w鈮�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

Measurements

Oscillations

Further Mechanics and Thermal Physics

Thermodynamics

Fundamentals of Physics

Translational Dynamics

Study anywhere. Anytime. Across all devices.

Let $f (t) = e^{i 蝇 t}$ on $(鈭� 蟺, 蟺)$ . Expand $f (t)$ in a complex exponential Fourier series of period $2 蟺$ . (Assume $w 鈮�$ integer.)