Compute the residue of a function

In summary, the conversation discusses a problem with a typo in an equation and how to solve it using Taylor series. It also mentions Morera's theorem and the Residue theorem. The suggestion is to use the Taylor series of ##1/(1-x)## to solve the problem.
  • #1
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Homework Statement
Find ##Res(h,z_0)##
Relevant Equations
##h=\frac{f}{g}##
final. problem 6.png

There is a typo. It should say ##h=\frac{f}{g}##.

Attempt: ##f## and ##g## are holomorphic on ##\Omega##. Homomorphic functions form a ##\mathcal{C}^*## algebra, so ##h## is holomorphic on ##\Omega## where ##g\neq 0##.

If ##z_0## is a removal singularity of ##h##, then ##Res(h,z_0)=0## by Morera's theorem.

Assume ##f(z_0)\neq 0##, then ##z_0## is a second order pole of h since ##g''(x_0)\neq 0 ##.
$$Res(h,z_0)=lim_{z\rightarrow z_0}\frac{d}{dz}\Big[\frac{f}{g}(z-z_0)^2\Big]$$
This method produces indeterminate forms, even after applying the L'Hopital's rule.

Let ##\mathcal{C}=\{z:|z-z_0|=1/2\}##. By Residue theorem,
$$Res(h,z_0)=\frac{1}{2\pi i}\int_\mathcal{C}\frac{f}{g}dz$$
This integral cannot be computed since ##f## and ##g## are not given.

Can anyone think of a clever method that could solve this problem?
 
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  • #2
You can just start writing out series. The Taylor series for f is known. For ##1/g##,, you know if looks like ##1/((z-z_0)^2(a+b(z-z_0)+c(z-z_0)^2+...) ## with a nonzero.

You can basically just bash this with the Taylor series of ##1/(1-x)##. Let ##x=\frac{1}{a}(-b(z-z_0)-c(z-z_0)^2+...)## and write
$$\frac{1}{g}= \frac{1}{(z-z_0)^2 a (1-x)}$$

Now apply the normal Taylor expansion to ##1/(1-x)##, and after the first couple terms you should get stuff that is high order in ##(z-z_0)## and hence can be ignored.
 
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  • #3
Office_Shredder said:
You can just start writing out series. The Taylor series for f is known. For ##1/g##,, you know if looks like ##1/((z-z_0)^2(a+b(z-z_0)+c(z-z_0)^2+...) ## with a nonzero.

You can basically just bash this with the Taylor series of ##1/(1-x)##. Let ##x=\frac{1}{a}(-b(z-z_0)-c(z-z_0)^2+...)## and write
$$\frac{1}{g}= \frac{1}{(z-z_0)^2 a (1-x)}$$

Now apply the normal Taylor expansion to ##1/(1-x)##, and after the first couple terms you should get stuff that is high order in ##(z-z_0)## and hence can be ignored.
Thank you so much for your reply. I will try this method.
 

Related to Compute the residue of a function

1. What is a residue in complex analysis?

A residue is a complex number that is obtained by evaluating a function at a point where the function is singular, or has a pole. It is an important concept in complex analysis as it helps to calculate integrals of functions with singularities.

2. How do you compute the residue of a function?

To compute the residue of a function, you first need to identify the singular points of the function, which are the points where the function is not defined or has a pole. Then, you can use the residue formula, which involves taking the limit of the function at the singular point and multiplying it by a certain factor. This factor depends on the type of pole at the singular point.

3. Why is it important to compute the residue of a function?

Computing the residue of a function is important because it allows us to evaluate complex integrals. This is because the residue theorem states that the integral of a function around a closed contour is equal to the sum of the residues of the function at its singular points inside the contour. This makes it a powerful tool in complex analysis and other areas of mathematics.

4. Can the residue of a function be negative?

Yes, the residue of a function can be negative. This can happen when the function has a pole of order greater than one at the singular point. In this case, the residue will be negative and its absolute value will be equal to the coefficient of the pole in the Laurent series expansion of the function.

5. Are there any applications of computing residues in real-world problems?

Yes, computing residues has many applications in various fields such as physics, engineering, and economics. For example, in physics, residues are used to calculate the residues of Green's functions, which are important in solving differential equations. In economics, residues are used to calculate the stability of economic systems. In general, residues are useful in solving problems that involve complex functions and integrals.

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