Elliptic functions, residue computation, same zeros and poles

In summary, the conversation is about understanding the concept of determining a constant in functions with the same zeros and poles. The person is confused about the method used to determine the constant and questions whether the residues are being compared.
  • #1
binbagsss
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Homework Statement



Hi,

I am trying to understand the attached:

gollygosh.png


I know that if two functions have zeros and poles at the same point and of the same order then they differ only by a multiplicative constant, so that is fine, as both have a double zero at ##z=w_j/2## and a double pole at ##z=0##.

But I don't understand at all the idea before determining what the constant ##C## should be?
I thought that perhaps we had set the residues at the double pole ##z=0## equal, but this is given by:

##\frac{1}{2}lim_{z \to 0} \frac{d}{dz}(z^2f(z)) ##,

whereas it looks like we've compared

##lim_{z \to 0} z^{2} f(z) ##,

so unless we have some reason to take the derivative outside the limit or something, I don't understand what we've done, and even whether my thoughts are on the right track and the residues are being compared?

Many thanks in advance.

Homework Equations



see above

The Attempt at a Solution



see above [/B]
 
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FAQ: Elliptic functions, residue computation, same zeros and poles

1. What are elliptic functions?

Elliptic functions are a special type of complex-valued function that are periodic in both the real and imaginary directions. They are used to describe the motion of a point on an elliptic curve and have applications in fields such as physics, engineering, and cryptography.

2. What is residue computation?

Residue computation is a mathematical technique used to evaluate the value of a complex function at a singular point. It involves finding the residue, or the coefficient of the term with the highest negative power, of the Laurent series expansion of the function at that point.

3. How are same zeros and poles related?

Same zeros and poles refer to the concept that a function can have the same number of zeros and poles within a given region of the complex plane. This is a property of certain types of functions, such as elliptic functions, and can have important implications in the study of their behavior.

4. What are the applications of elliptic functions in real-world problems?

Elliptic functions have a wide range of applications in various fields, including physics, engineering, and cryptography. They can be used to model physical systems, such as the motion of a pendulum, as well as to solve mathematical problems, such as computing the period of a function.

5. How are elliptic functions related to elliptic curves?

Elliptic functions and elliptic curves are closely related, as elliptic functions are used to describe the motion of a point on an elliptic curve. Elliptic curves are algebraic curves defined by a specific type of equation, and they have important applications in number theory and cryptography.

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