Residue Calculus for Evaluating e^z/cosh z on Circle |z|=5

In summary, residue calculus is a branch of complex analysis that utilizes the theory of residues to evaluate complex integrals. It involves expanding a function into a Laurent series and using the coefficients to calculate residues, which can then be used to evaluate the integral. This method can be applied to any contour, as long as the function being integrated is analytic within the contour. Some other applications of residue calculus include studying differential equations, number theory, and computing definite integrals, as well as its use in physics for evaluating scattering amplitudes in quantum field theory.
  • #1
David Laz
28
0
I need to use residue calculus to evaluate:
[tex]\oint_C {\frac{{e^z }}{{\cosh z}}} dz[/tex]
where C is the circle |z|=5
My only problem (which is a stupid one) is working out how many poles the function has inside the circle. I know its going to have them at pi*i/2 + pi*i*k. This is probably a really stupid question, but I've left a lot of this course till the last minute. :(
 
Physics news on Phys.org
  • #2
Worked it out. I'm an idiot for overlooking something so simple.
 

FAQ: Residue Calculus for Evaluating e^z/cosh z on Circle |z|=5

What is residue calculus?

Residue calculus is a branch of complex analysis that involves using the theory of residues (or residues at isolated singularities) to evaluate complex integrals. It is based on the concept that a function can be expanded into a Laurent series, and the coefficients of the series can be used to calculate the residues, which can then be used to evaluate the integral.

How is residue calculus used to evaluate e^z/cosh z on the circle |z|=5?

To evaluate e^z/cosh z on the circle |z|=5 using residue calculus, we first need to find the singular points of the function within the contour of the circle. In this case, the only singular point is at z=0, which is a pole of order 1. Next, we need to calculate the residue at z=0, which can be done using the formula Res(f,z0) = lim(z->z0) (z-z0)f(z). Finally, we can use the residue theorem to evaluate the integral, which states that the integral of a function over a closed contour is equal to 2πi times the sum of the residues of the function at its singular points within the contour.

What is the relationship between e^z and cosh z?

The relationship between e^z and cosh z is that cosh z is the real part of e^z. This means that cosh z is the hyperbolic cosine function, which is related to the exponential function e^z. In fact, cosh z can be expressed as (e^z + e^-z)/2, showing the connection between the two functions.

Can residue calculus be used for any contour?

Residue calculus can be used for any contour as long as the function being integrated is analytic (has no singularities) within the contour. However, it is most commonly used for integrals over closed contours, as the residue theorem applies specifically to these types of integrals.

What are some other applications of residue calculus?

Aside from evaluating complex integrals, residue calculus has many other applications in mathematics and physics. It is used in the study of differential equations, number theory, and in the computation of definite integrals. It also has applications in physics, such as in the evaluation of scattering amplitudes in quantum field theory.

Similar threads

Residue Theorem: Finding the Integral of z^3e^(-1/z^2) over |z|=5
Replies
6
Views
1K
Complex Integration using residue theorem
Replies
1
Views
2K
Find the residues of the following function + Cauchy Residue
Replies
2
Views
1K
I Calculate the residue of 1/Cosh(pi.z) at z = i/2 (complex analysis)
Replies
7
Views
2K
How Do You Evaluate sin(x)/x^4 Over the Unit Circle Using Complex Analysis?
Replies
3
Views
1K
What are the orders of the poles and the residue for sin(1/z)/cos(z)?
Replies
3
Views
3K
Contour integration around a complex pole
Replies
2
Views
2K
What Is the Residue of f(z) at z=0?
Replies
8
Views
2K
On pointwise convergence of Fourier series
Replies
3
Views
1K
Poles & Residues: Find & Order of z/cosz
Replies
12
Views
2K

Hot Threads

Recent Insights

Back
Top