Methods of Mathematical Physics-II

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Chapter#2: Fourier Series

Team Quanta gladly presents all short questions of Methods of Mathematical Physics – II’s Chapter#2: Fourier Series.

Q.1 Define Fourier series?

Answer:  A Fourier series is defined as the representation of a function in a series of sine  and cosine as;

    \[F\left(x\right)=\frac{a_0}{2}+\sum_{n=1}^{\infty}{a_n\cos{nx}+\sum_{n=1}^{\infty}{b_n\sin{nx}}}\]

Q.2 what is meant by in Fourier series?

Answer: The a_0,a_n ,and \ b_n are coefficients of Fourier series;

    \[a_0=\frac{1}{\pi}\int_{-\pi}^{+\pi}F\left(x\right)dx\]

    \[a_n=\frac{1}{\pi}\int_{-\pi}^{+\pi}{F\left(x\right)\cos{nx}dx}\]

    \[b_n=\frac{1}{\pi}\int_{-\pi}^{+\pi}{F\left(x\right)\sin{nx}dx}\]

Q.3 what is pathological?

Answer: Function which is not satisfying the condition is known as pathological.

Q.4 Describe Abel’s Theorem?

Answer: Abel’s theorem says that if a power series converges on (-1,+1) and also at x = 1 then its value at x = 1 is determined by continuity from the left of +1. You must know the series converges at x = 1 before you can apply Abel’ theorem.

Q.5 what is the use of Abel’s theorem in Fourier series?

Answer: Abel’s theorem allows us to conclude that if the Fourier coefficients F(n) = cn are known and F is piecewise continuous then F is determined.

Q.6 which theorem is called converse of Abel’s theorem in power series?

Answer: Converses to a theorem like Abel’s theorem are called Tauberian theorem. There is no exact converse but results conditional on some hypothesis. The field of divergent series and their summation methods contains many theorems of Abelian type and of Tauberian  type.

Q.7 To what value does the sum of Fourier series of F(x) converges at the point of continuity?

Answer: The sum of Fourier series of F(x) converges to the value F(a) at the continuous point.

Q.8 to what value does the sum of Fourier series of F(x) converges at the point of discontinuity?

Answer: At the discontinuity point x = a the sum of Fourier series of F(x) converges to;

    \[F\left(x_0\right)=\lim_{h\rightarrow 0}\left[\frac{F\left(x_0+h\right)+F(x_0-h)}{2}\right]\]

Q.9 If F\left(x\right)=x^2+x is expressed as a Fourier series in (-2,+2) to which value this series converges at x = 2 ?

Answer: The;

    \[F\left(x\right)=x^2+x\ \ \ ,\ \ \ \ \ -2\le x\le+2\]

Value to which the Fourier series of F(x) converges at x = 2 which is an end points are given  as;

    \[F\left(x\right)=\frac{f\left(-2\right)+f(+2)}{2}=\frac{\left(4-2\right)+(4+2)}{2}=4\]

The Fourier series converges at x = 2 to the value 4.

Q.10 Write down the properties of Fourier series?

Answer: The properties of Fourier series are given as;

  • F(x) is continuous at -\pi\le x\le\pi
  • F(-x)=F(x)
  • F^\prime\left(x\right) is sectionally continuous.

Q.11 If is defined -\pi<x<+\pi in write the value of a0 , an  ?

Answer: Given as;

    \[F\left(x\right)=\sin{hx}\]

    \[F\left(-x\right)=\ \sin{h\left(-x\right)}\]

    \[F\left(-x\right)=-\sin{hx}\]

    \[=-F\left(x\right)\]

\sin{hx} is an odd function. The value of a_0=0\ and\ a_n=0.

Q.12 Write down the formula of Fourier series constant for F(x) in the interval (- p,+p)?

Answer: The Fourier constants for F(x) in the interval (-p,+p) are given as;

    \[a_0=\frac{1}{\pi}\int_{-\pi}^{+\pi}F\left(x\right)dx\]

    \[a_n=\frac{1}{\pi}\int_{-\pi}^{+\pi}{F\left(x\right)\cos{nx}dx}\]

    \[b_n=\frac{1}{\pi}\int_{-\pi}^{+\pi}{F\left(x\right)\sin{nx}dx}\]

Q.13 Find the constant a0 of the Fourier series for function F(x) = x in 0 \le x \le 2\pi

Answer: Given that;

    \[F\left(x\right)=x\]

    \[a_0=\frac{1}{\pi}\int_{-\pi}^{+\pi}F\left(x\right)dx\]

    \[a_0=\frac{1}{\pi}\int_{0}^{+2\pi}xdx=\frac{1}{\pi}(\frac{x^2}{2})\]

    \[a_0=\frac{1}{\pi}\left[\frac{(2\pi)^2-0}{2}\right]\]

    \[a_0=\frac{1}{\pi}\left[\frac{4\pi^2}{2}\right]=2\pi\\]

Q.14 If F\left(x\right)=\left|x\right| expanded as a Fourier series in \left(-\pi,+\pi\right) Find ao?

Answer: The given function is an even function;

    \[a_0=\frac{1}{\pi}\int_{-\pi}^{+\pi}F\left(x\right)dx\]

    \[a_0=\frac{1}{\pi}\int_{-\pi}^{+\pi}\left|x\right|dx=\frac{2}{\pi}\int_{0}^{+\pi}xdx\]

    \[a_0=\frac{2}{\pi}\left(\frac{x^2}{2}\right)=\pi\]

Q.15 Find the Fourier series coefficients of a0 of

Answer: Solution is;

    \[a_0=\frac{1}{\pi}\int_{-\pi}^{+\pi}F\left(x\right)dx\]

    \[a_0=\frac{1}{\pi}\int_{-\pi}^{+\pi}{e^xdx}\]

    \[a_0=\frac{1}{\pi}e^x=\frac{1}{\pi}\left(e^{+\pi}-e^{-\pi}\right)\]

Q.16 Find bn in the expansion of x2 as Fourier series in (-p,+p)?

Answer: The; F\left(x\right)=x^2 is even function. So the value of b_n=0

    \[a_0=\frac{2\sin{h\pi}}{\pi}\]

Q.17 State the convergence condition on Fourier series?

Answer: The Fourier series of F(x) converges to F(x) at all points where F(x) is continuous.

At a point of discontinuity x0, the series converges to the average of the left limit and right limit of F(x) at x0

    \[F\left(x_0\right)=\frac{1}{2}\left[\lim_{h\rightarrow 0}{F\left(x_0+h\right)+\lim_{h\rightarrow 0}{F(x_0-h)}}\right]\]

Q.18 Write the complex form of Fourier series for F(x) defined in the interval  (C,C+2l)  ?

Answer: The series for F(x) defined in the interval (C,C+2l) and satisfying Dirichlet’s conditions can be given in the form of;

    \[F\left(x\right)=\sum_{N=-\infty}^{+\infty}{C_ne^{inx}}\]

Where

    \[C_n=\frac{1}{2\pi}\int_{C}^{C+2\pi}{F(x)e^{-inx}dx}\]

Q.19 what do you mean by Harmonic analysis?

Answer: The process of finding the Fourier series of the periodic function y=F(x) of period 2l or 2p using the numerical values of x and y’ is known as Harmonic analysis.

Q.21 what is meant by periodic Sturm-lioville equation?

Answer: On the interval $\ a\le x\le b$ satisfying the boundary condition of the form;

    \[k_1y\left(a\right)+k_2y^\prime\left(a\right)=0\]

    \[l_1y\left(b\right)+l_2y^\prime\left(b\right)=0\]

Is called periodic Sturm-lioville equation

Q.22 Write the properties of linear operator?

Answer: The properties are;

The operator is linear because it satisfy the condition;

    \[\mathcal{L}\left[ay_1\left(x\right)+by_2(x)\right]=a\mathcal{L}y_1\left(x\right)+b\mathcal{L}y_2(x)\]

The Sturm-lioville operator is said to be self adjoint on the interval [a,b] if it satisfy.

    \[\int_{a}^{b}{\mathrm{\Psi}\left(\mathcal{L}\emptyset\right)dx=\int_{a}^{b}{\emptyset(\mathcal{L}\mathrm{\Psi})^\ }}dx\\]

Where are real functions

The operator is Hermite on the interval [a,b] then

    \[\int_{a}^{b}{\mathrm{\Psi}^\ast\left(\mathcal{L}\emptyset\right)dx=\int_{a}^{b}{(\mathcal{L}\mathrm{\Psi})^\ast}}\emptyset dx\]

Q.23 what are the properties of Sturm-lioville equation?

Answer: There are three properties of Sturm-lioville equation;

It contains linear differential operator of 2nd order;

    \[\mathcal{L}=\frac{d}{dx}\left[p(x)\frac{d}{dx}\right]+q(x)\]

The sign of λ is non-negative and λ is real.

Solution y(x) is banded.

Q.24 Write down the examples of Sturm-lioville equation?

Answer: The examples of Sturm-lioville equation are given as;

  • Lagendre equation
  • Laguerr’s equation
  • Hermite equation
  • Bessel’s equation.

Q.26 what is the purpose of non-periodic function in Fourier series?

Answer: Fourier representation may be used as non-periodic function. If we wish to find the Fourier series of non-periodic function with in fixed range, then we may continue this function outside this range, so as to make it periodic. The Fourier transformation of this periodic function would them correctly replacing the non-periodic function in the desired   range. We may also extend the function at liberty to make it odd or even. For odd an=0 and for even bn=0.

Q.27 what is orthogonal?

Answer: Two vectors are orthogonal if their inner product is zero. In other words,

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