Solving the Pole and Residue at a for f

In summary, the conversation discusses a problem involving analytic functions and the identification of a pole and corresponding residue. The approach of using Laurent expansion is initially considered but deemed too complicated, and Taylor expansion is then attempted with some difficulties. The conversation also includes questions about the use of Taylor expansion for complex functions and the possibility of ignoring negative terms. The proposed solution involves proving that h(z) can be written as z multiplied by an analytic function, and the corresponding value of this function at a is discussed.
  • #1
latentcorpse
1,444
0
Im getting really bugged by this question:

Let g and h be analytic in the open disc [itex]\{z \in \mathbb{C} : |z-a| < r \}[/itex], r>0and let [itex]f(z)=\frac{g(z)}{h(z)}[/itex]

if [itex]g(a) \neq 0, h(a)=0, h'(a) \neq 0[/itex] show that f has a pole at z=a and find the corresponding residue of f at a.


Now I initially though we had to Laurent expand the functions g and h but that got really complicated very quickly and i couldn't find anyway of sorting stuff out.
then i tried taylor expanding them which was nicer but still didn't rearrange well.

so my 2 questions are:
(i) how do i do the above problem, and
(ii) am i correct in saying that you can't taylor expand a complex function you can only laurent expand it and if the negative coefficients turn out to be 0 then it reduces to a taylor expansion or are we allowed to do taylor expansions? if we are allowed to, aren't we ignoring the negative terms - is this allowed?
 
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  • #2
Try proving first h(z)=zq(z) for some analytic q s.t. q(a) neq 0. [Here q(a)=??]
 
  • #3
q(a)=g(a)/(a f(a))? why are we doing this though?
 

What is the purpose of solving the Pole and Residue at a for f?

The Pole and Residue at a for f is a mathematical technique used to solve complex integrals in the field of complex analysis. It allows us to calculate the values of integrals that would otherwise be difficult or impossible to obtain using traditional methods.

How is the Pole and Residue at a for f calculated?

The Pole and Residue at a for f is calculated by finding the singularities (poles) of the complex function f and then evaluating the residues at those poles. The residues are then added together to obtain the final result.

What is a pole in complex analysis?

In complex analysis, a pole is a point where a complex function becomes infinite or undefined. It is usually represented by a singularity in the complex plane. Poles are important in the study of complex functions as they can greatly affect the behavior and properties of the function.

What are the applications of solving the Pole and Residue at a for f?

The Pole and Residue at a for f has many applications in physics, engineering, and mathematics. It is used to solve integrals in electromagnetic theory, quantum mechanics, and fluid dynamics, among others. It is also used in the analysis of signals and systems in control theory and signal processing.

What are some tips for solving the Pole and Residue at a for f effectively?

Some tips for solving the Pole and Residue at a for f effectively include understanding the concept of poles and residues, being familiar with complex functions and their properties, and practicing with various examples. It is also important to carefully evaluate the residues and to check for any mistakes in the calculations.

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