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Chapter 14: Problem 10

The values of the following integrals are known and can be found in integral tables or by computer. Your goal in evaluating them is to learn about contour integration by applying the methods discussed in the examples above. Then check your answers by computer. $$\int_{-\infty}^{\infty} \frac{d x}{x^{2}+4 x+5}$$

Short Answer

Expert verified
The value of the given integral is \(\pi\).

Step by step solution

01

- Identify the integrand

The given integral is \(\int_{-\infty}^{\infty} \frac{dx}{x^{2}+4x+5}\). Recognize that this is a rational function, which often suggests contour integration as a method.
02

- Complete the square

Rewrite the quadratic in the denominator by completing the square: \(x^{2} + 4x + 5 = (x + 2)^2 + 1\). Hence, the integral becomes \(\int_{-\infty}^{\infty} \frac{dx}{(x + 2)^2 + 1}\).
03

- Setup the contour integral

To use contour integration, consider the complex function \(f(z) = \frac{1}{(z+2)^2 + 1}\). This function has poles at \(z = -2 + i\) and \(z = -2 - i\).
04

- Choose a semi-circular contour

Choose a semi-circular contour in the upper half-plane that closes on the real axis. This will allow us to evaluate the integral using the residue theorem.
05

- Apply the residue theorem

By the residue theorem, the integral around a closed contour that includes the pole \(z = -2 + i\) is \(2\pi i \) times the residue at that pole. Compute the residue at \(z = -2 + i\):
06

- Compute the residue

The residue at \(z = -2 + i\) can be found as follows: \(\text{Res}(f, -2 + i) = \lim_{z \to -2 + i} (z - (-2 + i)) \frac{1}{(z + 2)^2 + 1} = \frac{1}{2i}\).
07

- Sum up residues

Since the integral along the real axis from \(-\infty \) to \(+\infty \) is twice the integral along the contour (from symmetry and the fact that the integral over the arc vanishes as its radius goes to \(\infty\)), the value of the original integral is \(2\pi i \cdot \frac{1}{2i} = \pi\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Residue Theorem
The residue theorem is a powerful tool in complex analysis used to evaluate contour integrals. It relates the complex integral around a closed contour to the sum of residues within that contour.
A residue is a special value calculated at the poles of a complex function. Poles are points where the function goes to infinity.
To use the residue theorem:
  • Identify the poles inside the contour.
  • Calculate the residues at these poles.
  • Sum the residues.
The integral around the contour is then given by multiplying this sum by \(2\pi i\).
This method simplifies the evaluation of complex integrals significantly.
Complex Functions
Complex functions are functions that take and return complex numbers. They can be expressed in the form \(f(z) = u(x, y) + iv(x, y)\), where \(z = x + iy\).
These functions are essential in various fields like fluid dynamics, electromagnetism, and quantum mechanics.
Some properties of complex functions include:
  • Analyticity: A function is analytic (or holomorphic) if it is differentiable at every point in its domain.
  • Poles: Points where the function takes an infinite value.
  • Zeros: Points where the function equals zero.
Understanding these properties helps in contour integration and applying the residue theorem efficiently.
Integral Evaluation
Evaluating integrals, especially with complex functions, often requires special techniques.
Contour integration is one such technique that simplifies the process by converting a real integral into a complex one. Here's how:
  • Identify the integrand and express it in terms of a complex function.
  • Choose an appropriate contour for integration (like a semicircle in the upper half-plane).
  • Use the residue theorem to evaluate the integral around the contour.
For example, consider the integral \( \int_{-\infty}^{\infty} \frac{dx}{x^{2}+4x+5}\). By rewriting the quadratic in the denominator, it becomes easier to apply contour integration and find the result as shown in the problem solution.

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Most popular questions from this chapter

If \(w=f(z)=u(x, y)+i v(x, y), f(z)\) analytic, defines a transformation from the variables \(x, y\) to the variables \(u, v,\) show that the Jacobian of the transformation (Chapter 5, Section 4) is \(\partial(u, v) / \partial(x, y)=\left|f^{\prime}(z)\right|^{2} .\) Hint. To simplify the determinant, use the Cauchy-Riemann equations and the equations (Section 2) used in obtaining them.

Show that the following functions are harmonic, that is, that they satisfy Laplace's equation, and find for each a function \(f(z)\) of which the given function is the real part. Show that the function \(v(x, y)\) (which you find) also satisfies Laplace's equation. $$3 x^{2} y-y^{3}$$

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The fundamental theorem of algebra says that every equation of the form \(f(z)=\) \(a_{n} z^{n}+a_{n-1} z^{n-1}+\cdots+a_{0}=0, a_{n} \neq 0, n \geq 1,\) has at least one root. From this it follows that an \(n\) th degree equation has \(n\) roots. Prove this by using the argument principle. Hint: Follow the increase in the angle of \(f(z)\) around a very large circle \(z=r e^{i \theta} ;\) for sufficiently large \(r,\) all roots are inclosed, and \(f(z)\) is approximately \(\overline{a_{n}} z^{n}\)
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