Complex Analysis: Sums of elementary fractions

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SUMMARY

The discussion focuses on the mathematical process of representing the rational function f(z) = (z-2)/(z^2+1) as sums of elementary fractions and finding its primitive functions (indefinite integrals). The user seeks clarification on the holomorphic nature of the function, emphasizing that while the integral of f(z) may be complex, it can be simplified into easier components: ∫(z/(z^2+1)) dz and -2∫(1/(z^2+1)) dz. This breakdown facilitates the integration process by leveraging simpler integrals.

PREREQUISITES
  • Understanding of rational functions and their properties
  • Knowledge of holomorphic functions in complex analysis
  • Familiarity with integration techniques, specifically indefinite integrals
  • Experience with elementary fractions and partial fraction decomposition
NEXT STEPS
  • Study the method of partial fraction decomposition in complex analysis
  • Learn about holomorphic functions and their properties
  • Explore techniques for integrating rational functions
  • Investigate the application of residues in complex integration
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Students and professionals in mathematics, particularly those studying complex analysis, integration techniques, and rational functions. This discussion is beneficial for anyone looking to deepen their understanding of holomorphic functions and integration strategies.

Potage11
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I have a homework question that reads:
Represent the following rational functions as sums of elementary fractions and find the primitive functions ( indefinite integrals );

(a) f(z)=z-2/z^2+1

But my confusion arrises when I read sums of elementary fractions.
I think what the question is asking is, show that it is holomorphic, in order to use the property g'(z)=f(z).

Could someone clarify this maybe?
 
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It just means that as a whole, [tex]\int \frac{z-2}{z^2+1} dz[/tex] may be a hard integral, but [tex]\int \frac{z}{z^2+1} dz -2\int \frac{1}{z^2+1} dz[/tex] are 2 easy ones.
 

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