# Integrate sin(bx)/sinh(ax) from zero to infinity

Not what you're looking for?

Do the integral (0 --> infinity) using contour integration:

J(a,b) = dx sin(bx)/sinh(ax)

##### Purchase this Solution

##### Solution Summary

We explain in detail using the methods of complex analysis, how to evaluate the integral.

##### Solution Preview

We note that the integrand is an even function of x, so we can write:

J(a,b) = 1/2 Integral from minus to plus infinity of sin(bx)/sinh(ax) dx

It is convenent to scale out eiter of the constants a or b. Let's get rid of a by putting ax = u:

J(a,b) = 1/(2a) Integral from minus to plus infinity of sin(b/a u)/sinh(u) du

If we put k = b/a, then we can write:

Integral from minus to plus infinity of sin(k x)/sinh(x) dx =

Integral from minus to plus infinity of Im[exp(i k x)]/sinh(x) dx =

Limit epsilon to zero of

Integral from minus infinity to minus epsilon and then from epsilon to plus infinity of Im[exp(i k x)]/sinh(x) dx =

Limit epsilon to zero of

Im [ Integral from minus infinity to minus epsilon and then from epsilon to plus infinity of exp(i k x)/sinh(x) dx ] =

Limit epsilon to zero and R to infinity of

Im [ Integral from minus R to minus epsilon and then from epsilon to R of exp(i k x)/sinh(x) dx ] (1)

To evaluate this, we first assume k that k>=0 and consider the contour integral of exp(i k z)/sinh(z) from -R to -epsilon, and then along a counterclockwise semi-circle of radius epsilon with center the origin, we travel from minus epsilon to epsilon, and then from epsilon we go to R, and then along a counterclockwise semi-circle of radius R with center the origin, we travel back from R to minus R.

Then this contour integral includes ...

##### Purchase this Solution

##### Free BrainMass Quizzes

##### Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

##### Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

##### Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

##### Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

##### Probability Quiz

Some questions on probability