Chapter 7: Q13P (page 360)

If
$f (x) = \frac{1}{2} a_{0} + \sum_{1}^{\infty} a_{n} c o s n x + \sum_{1}^{\infty} b_{n} s i n n x = \sum_{- \infty}^{\infty} c_{n} e^{i n x}$ , use Euler's formula to find $a_{n}$ and $b_{n}$ in terms of $c_{n}$ and $- c_{n}$ , and to find $c_{n}$ and $- c_{n}$ in terms of $a_{n}$ and $b_{n}$ a.

Short Answer

Expert verified

The resultant expansion $c_{n}$ and $c_{- n}$ are $c_{n} = \frac{1}{2 i} (i a_{n} + b_{n})$ and $c_{- n} = \frac{1}{2} (a_{n} + i b_{n})$ .

Step by step solution

Given data

The given function is $f (x) = \frac{1}{2} a_{0} + \sum_{1}^{\infty} a_{n} \cos n x + \sum_{1}^{\infty} b_{n} \sin n x = \sum_{- \infty}^{\infty} c_{n} e^{i n x}$ .

Concept of Fourier series

An infinite sum of sines and cosines is used to represent the expansion of a periodic function f(x) into a Fourier series.

The orthogonality relationships of the sine and cosine functions are used in Fourier series.

Simplify the expression

By Fourier series expansion, $f (x) = \sum_{n = - \infty}^{\infty} c_{n} e^{i n x}$ and $f (x) = c_{0} + \sum_{n = - \infty}^{- 1} c_{n} e^{i n x} + \sum_{n = 1}^{\infty} c_{n} e^{i n x}$ .

Check and simplify the expression

For check, change $n \to - n$ .

$f (x) = c_{0} + \sum_{n = 1}^{\infty} [c_{- n} e^{- i n x} + c_{n} e^{i n x}] f (x) = c_{0} + \sum_{n = 1}^{\infty} [(c_{n} + c_{- n}) \cos (n x) + i (c_{n} + c_{- n}) \sin n x]$

By comparison of above series:

$a_{0} = 2 c_{0} a_{n} = c_{n} + c_{- n} b_{n} = i (c_{n} + c_{- n})$

This implies that $c_{0} = \frac{a_{0}}{2}$ .

$c_{n} = \frac{1}{2 i} (i a_{n} + b_{n}) c_{- n} = \frac{1}{2} (a_{n} + i b_{n})$

Therefore, the $c_{n}$ and $c_{- n}$ are $c_{n} = \frac{1}{2 i} (i a_{n} + b_{n})$ and $c_{- n} = \frac{1}{2} (a_{n} + i b_{n})$ respectively.

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If
$f (x) = \frac{1}{2} a_{0} + \sum_{1}^{\infty} a_{n} c o s n x + \sum_{1}^{\infty} b_{n} s i n n x = \sum_{- \infty}^{\infty} c_{n} e^{i n x}$ , use Euler's formula to find $a_{n}$ and $b_{n}$ in terms of $c_{n}$ and $- c_{n}$ , and to find $c_{n}$ and $- c_{n}$ in terms of $a_{n}$ and $b_{n}$ a.

Short Answer

Step by step solution

Given data

Concept of Fourier series

Simplify the expression

Check and simplify the expression

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Iff(x)=12a0+∑1∞ancosnx+∑1∞bnsinnx=∑-∞∞cneinx, use Euler's formula to find an and bnin terms of cnand -cn, and to find cnand -cnin terms of anandbn a.

Short Answer

Step by step solution

Given data

Concept of Fourier series

Simplify the expression

Check and simplify the expression

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Classical Mechanics

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Circular Motion and Gravitation

Conservation of Energy and Momentum

Study anywhere. Anytime. Across all devices.