Partial fraction decomposition with complex function

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SUMMARY

The discussion centers on the challenge of performing partial fraction decomposition on the expression e^(icx)/(x^2 + a^2)^2, where a and c are positive constants. Participants highlight that the desired decomposition into the form A/(x^2 + a^2) + B/(x^2 + a^2) is not feasible due to the transcendental nature of e^(icx), which cannot be expressed as a sum of rational functions. Instead, it is suggested to utilize Laplace transforms to manipulate the expression effectively.

PREREQUISITES
  • Understanding of partial fraction decomposition
  • Familiarity with Laplace transforms
  • Knowledge of transcendental functions
  • Basic calculus concepts related to complex functions
NEXT STEPS
  • Research how to apply Laplace transforms to complex functions
  • Study the properties of transcendental functions in mathematical analysis
  • Explore advanced techniques in partial fraction decomposition
  • Learn about the implications of rational vs. transcendental functions in calculus
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Mathematicians, engineering students, and anyone involved in complex analysis or signal processing who needs to understand the limitations of partial fraction decomposition with transcendental functions.

thayin
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As part of a project I have been working on I fin it necessary to manipulate the following expression.

e^(icx)/(x^2 + a^2)^2 for a,c > 0

I would like to decomp it into the form

A/(x^2 + a^2) + B/(x^2 + a^2) = e^(icx)/(x^2 + a^2)^2

but I am having trouble getting a usable outcome.
 
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thayin said:
As part of a project I have been working on I fin it necessary to manipulate the following expression.

e^(icx)/(x^2 + a^2)^2 for a,c > 0

I would like to decomp it into the form

A/(x^2 + a^2) + B/(x^2 + a^2) = e^(icx)/(x^2 + a^2)^2

but I am having trouble getting a usable outcome.

You need to apply laplace's transform to this to solve for this. you have the write set up, BUT

A/(x^2+a^2) + B/((x^2+a^2)^2), look up laplace transforms and do it as so... you can solve it then.

Yus310
 
thayin said:
As part of a project I have been working on I fin it necessary to manipulate the following expression.

e^(icx)/(x^2 + a^2)^2 for a,c > 0

I would like to decomp it into the form

A/(x^2 + a^2) + B/(x^2 + a^2) = e^(icx)/(x^2 + a^2)^2

but I am having trouble getting a usable outcome.


No wonder you're having problems: it can't be done, as your function is a transcendental function (i.e., e^{icx} is NOT a polynomial, whereas the sum of fractions you want is a rational function.

DonAntonio
 

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