User:Egm6322.s12.TEAM1.yang.HW1

Problem 2: Fourier Coefficients
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Find
A. Verify Eq 2.1 is valid. B. Recover from Eq2.2 the Fourier coefficients found with the trigonometric identities given.

Solution
A. Verify Eq 2.1 is valid.

if $$ n=m$$, then:

if $$ n\ne m$$, then:

B. Recover from Eq2.2 the Fourier coefficients found with the trigonometric identities given.

From the definition of the coefficients in the lecture, we can get: Then, the Fourier coefficients can be found from above:

Problem 3: Alternative Coefficients
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Given & Find
show that, equivalent to (2)-(3)P.42-10, the Fourier series can be written as

$$f(\theta )=\operatorname{Re}(\sum\limits_{n=0}^{+\infty }{{e}^{in\omega \;\theta \;}})$$

An advantage of (2)-(3)P.42-10 is a simple and elegant computation of the coefficients as shown in (1) p.42-12.

Solution
Starting from eqation (1) in P42-10, the following can be derived.

in which:

$${{\overline{a}}_{n}}={{a}_{n}}-i{{b}_{n}}$$, for n>0.

In comparison, the coefficients in equation (1) p.42-12 is obviously more simple and elegant than equation(3.1) above.

Problem 7: Computing Coefficients
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Given
$$q(x)=-4-7x+3{{x}^{2}}+2{{x}^{3}}-5{{x}^{4}}+{{x}^{5}}$$

Find
Find $${{d}_{i}}$$ such that: $$q(x)=\sum\limits_{i=0}^{n}{{{d}_{i}}{{P}_{i}}(x)}\in {{\Rho }_{n}}$$ plot q(x) in two figures

Solution
According the Lecture 42-15, components can be computed by:

And the Legendre polynomials are known as: And, The components can be computed by substituting Eq 7.2 and Eq 7.3 into Eq 7.1.

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