Solving the Integral Using Contour Integration: A Comprehensive Guide

In this article, we will explore the process of solving the integral involving the square root function using contour integration techniques. Specifically, we will examine the integral:

I int_{0}^{infty} frac{sqrt{x}}{1 x^3} dx

Introduction to the Integral and Branch Cuts

The integrand frac{sqrt{x}}{1 x^3} includes a square root function, which introduces a branch cut. We will define the branch cut along the negative real axis for the square root function. This will allow us to properly handle the multivalued nature of the square root function.

Step-by-Step Solution Using Contour Integration

Step 1: Identify the Function and Branch Cut

First, we recognize that the integrand involves the square root function, which requires a branch cut. We define the branch cut of the square root function along the negative real axis.

Step 2: Consider the Contour

To evaluate the integral, we consider a contour in the complex plane that consists of the following parts:

The line segment along the positive real axis from 0 to R. A semicircular arc from R to -R in the upper half-plane. The line segment along the negative real axis from -R to 0.

As R to infty, the integral over the semicircular arc will vanish due to the decay of the integrand.

Step 3: Parametrize the Integral

On the positive real axis, we set z x and thus sqrt{z} sqrt{x}. On the negative real axis, we set z -x leading to sqrt{z} isqrt{x}.

Step 4: Evaluate the Integral Along the Contour

The integral around the closed contour can be expressed as:

oint_C frac{sqrt{z}}{1 z^3} dz int_0^R frac{sqrt{x}}{1 x^3} dx int_{text{arc}} int_{-R}^0 frac{isqrt{-x}}{1 - x^3} dx

Step 5: Analyze the Poles

The function frac{1}{1 z^3} has poles at the cube roots of -1:

z_k e^{ifrac{2k 1pi}{3}} quad k 0, 1, 2

Only the pole in the upper half-plane, z_0 e^{ifrac{pi}{3}}, will contribute to the residue.

Step 6: Calculate the Residue

The residue at the pole z_0 can be computed as follows:

text{Residue} left(frac{sqrt{z}}{1 z^3} quad z_0right) lim_{z to z_0} (z - z_0) frac{sqrt{z}}{1 z^3}

Step 7: Use Residue Theorem

By the residue theorem:

oint_C frac{sqrt{z}}{1 z^3} dz 2pi i times text{Residue}

Step 8: Combine Results

The integral over the entire contour will equal the sum of the integrals along the positive and negative axes and the integral over the arc. As R to infty, the arc integral vanishes. The contributions from the positive and negative parts will be related by sqrt{-x} isqrt{x}.

Step 9: Final Calculation

After evaluating and simplifying, we find:

I int_0^{infty} frac{sqrt{x}}{1 x^3} dx frac{pi}{3sqrt{3}}

Thus, the integral can indeed be solved using contour integration with careful consideration of branch cuts.