Integrating Exponential and Trigonometric Functions Using Complex Analysis and Gauss’ Mean-Value Theorem

In this article, we explore the integration of a complex function involving trigonometric and exponential components. Specifically, we evaluate the integral:

Introduction

The integral in question is:

I ∫02π ecosthetacos(sinθ)dθ

Complex Analysis Approach

To evaluate this integral, we will use techniques from complex analysis. First, we recognize that the integrand can be expressed in terms of complex exponentials. This is a common technique in complex analysis to simplify integrals involving trigonometric functions.

Step 1: Express the Integrand Using Complex Exponential Functions

Notice that:

cos(sinθ) Re(eisinθ)

Thus, the integral can be rewritten as:

I Re(∫02π ecostheta eisinθdθ)

Step 2: Combine the Exponents

We combine the exponents to get:

ecosthetaisinθ eethniceta

This simplifies the integral to:

I Re(∫02π eethnicetadθ)

Step 3: Evaluate the Integral

The integral:

∫02π eethnicetadθ

is a standard result in complex analysis and equals:

2πI_01

where I_01 is the modified Bessel function of the first kind of order zero.

Step 4: Find the Real Part

Hence, we have:

I Re(2πI_01)

Since I_01 is a real number, we can conclude:

I 2πI_01

Physicist's Approach Using Gauss' Mean-Value Theorem

From a physicist's perspective, we can also evaluate the integral using Gauss' mean-value theorem. This theorem tells us that the average value of an analytic function on a closed disk is equal to the value of the function at its center.

Step 1: Understanding the Theorem

Gauss' mean-value theorem states that for a function g(z) with a convergent power series inside a closed disk of radii R enclosed by circle with center s:

?g(z)?_ 1/2π ∫02π dθ · g(s Reitheta) g(s)

Step 2: Applying the Theorem to Our Integral

Here, we are evaluating the average value of ez on an origin-centered unit circle, so:

s 0

R 1

Thus, according to the theorem:

?ez 1/2π ∫02π ezdθ 1/2π ∫02π exp(cos(θ) isin(θ))dθ 1/2π ∫02π (exp(costheta)cos(sinθ) exp(costheta)isin(sinθ))dθ

Step 3: Simplifying the Expression

We can split the expression into real and imaginary parts:

1/2π ∫02π exp(costheta)cos(sinθ)dθ 1/2π ∫02π exp(costheta)isin(sinθ)dθ exp(0) 1

The real part of the expression is:

1/2π ∫02π exp(costheta)cos(sinθ)dθ 1

Final Result

The value of the integral is:

∫02π ecosthetacos(sinθ)dθ 2πI_01

Where I_01 ≈ 1.266065877752008, and hence:

I ≈ 2π(1.266065877752008) ≈ 7.9587

Thus, the final answer is:

∫02π ecosthetacos(sinθ)dθ ≈ 7.9587