Show that ∫0^2 π e^con θ cos(n ϑ-sinϑ) d ϑ=(2 π)/(n!) |...

Name: Problem 4, Chapter 4: Methods of Theoretical Physics, Part I
Uploaded: 2022-01-19T22:33:10-08:00
Duration: 6 min 6 s
Channel: Eduard Sanchez
Description: Show that ∫0^2 π e^con θ cos(n ϑ-sinϑ) d ϑ=(2 π)/(n!)

Key Concepts

Jacobi-Anger Expansion

The Jacobi-Anger expansion is a method for expressing exponentials of trigonometric functions as infinite series involving Bessel functions. This expansion is particularly useful when dealing with integrals that include terms like exp(?i sin?) or cos(n? ? sin?), since it converts the oscillatory behavior into a sum of simpler, well-studied functions.

Orthogonality in Fourier Analysis

Fourier analysis relies on the property that trigonometric functions, like sine and cosine, are orthogonal over a complete period. This orthogonality simplifies the evaluation of integrals involving products of trigonometric functions, as it allows isolation of individual terms in a series expansion and helps in deducing the contributions that survive the integration over the entire interval.

Generating Functions

Generating functions provide a systematic way of encoding an infinite sequence of coefficients (such as those in the series representation of exponential functions or Bessel functions) into a single function. These functions are instrumental in converting a complex integral into an expression that reveals a pattern or identity, such as the one involving factorial terms.

Complex Exponential Representation

This concept involves rewriting trigonometric functions using Euler's formula, which expresses cosine and sine in terms of complex exponentials. By representing a cosine function as the real part of an exponential, one can simplify the integral into a form more amenable to series expansion or integration, particularly when combined with other exponential terms.

Bessel Functions

Bessel functions naturally arise in problems exhibiting cylindrical symmetry and in the series representations of functions involving exponentials of trigonometric functions. They serve as the coefficients in the Jacobi-Anger expansion, thereby linking the integrand to known properties and integrals of Bessel functions which can lead to compact closed-form results.

Transcript

0:00 Hi there.

00:01 So for this problem, by considering that 1 plus the exponential of i times theta and all of these elevated to the n, this expression right here, we need to use that to prove these relations right here, where we are also given that the constant cn, the coefficient cn, is equal to n factorial divided by r factorial and this times n minus r factorial.

00:50 So first we try to express that 1 plus the exponential of i -teta in terms of its real and imaginary pars.

01:05 So for this we use euler's equation and then the half angle formula.

01:11 So we start that by 1 plus the exponential of i times theta is equal to 1 plus the cosine of theta, and this plus i times sine of theta.

01:26 And then using the half angle formula, we know that this is equal to 2 times the cosine square of half of theta, and this plus plus i times 2 the sign of theta divided by 2 and this times cosine of theta divided by 2.

02:00 Thus, if we elevate this to the end, we will have the following.

02:09 We will have that this is just simply all of this plus i 2, sign of half of theta, cosine of half of theta, and all of these elevated to the n.

02:28 Now, from this, we obtained that this is equal to 2 elevated to the n, and this cosine of theta divided by 2, elevated to the n, and this the exponential of i times theta divided by 2, and this elevated to the n.

02:48 But we also have from the binomial expansion, from the binomial expansion, we know that 1 plus the esponemtial of i times theta should be equal to 1 plus n times the esponemcial of theta.

03:07 Dot, dot, dot, dot.

03:09 And this will give us to the nth term that is going to be the following...

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