Partial Fractions in Complex Analysis - Calculation


Let f(z) be a function meromorphic in the finite complex plane with poles at λ1, λ2, ..., and let (Γ1, Γ2, ...) be a sequence of simple closed curves such that:

  • The origin lies inside each curve Γk
  • No curve passes through a pole of f
  • Γk lies inside Γk+1 for all k
  • , where d(Γk) gives the distance from the curve to the origin

Suppose also that there exists an integer p such that

Writing PP(f(z); z = λk) for the principal part of the Laurent expansion of f about the point λk, we have

if p = -1, and if p > -1,

where the coefficients cj,k are given by

λ0 should be set to 0, because even if f(z) itself does not have a pole at 0, the residues of f(z)/zj+1 at z = 0 must still be included in the sum.

Note that in the case of λ0 = 0, we can use the Laurent expansion of f(z) about the origin to get

so that the polynomial terms contributed are exactly the regular part of the Laurent series up to zp.

For the other poles λk where k ≥ 1, 1/zj+1 can be pulled out of the residue calculations:

To avoid issues with convergence, the poles should be ordered so that if λk is inside Γn, then λj is also inside Γn for all j < k.

Read more about this topic:  Partial Fractions In Complex Analysis

Other articles related to "calculation":

Measuring Seasonality - Calculation - Remarks
2 ... In a multiplicative time-series model, the seasonal component is expressed in terms of ratio and percentage as Seasonal effect = (T*S*C*I)/( T*C*I)*100 = Y/(T*C*I )*100 However in practice the detrending of time-series is done to arrive at S*C*I ...
Liu Hui's π Algorithm - Quick Method
... Calculation of square roots of irrational numbers was not an easy task in the third century with counting rods ... quite happy with this result because he had acquired the same result with the calculation for a 1536-gon, obtaining the area of a 3072-gon ... After all calculation of square roots was not a simple task with rod calculus ...
Calculation - Comparison To Computation
... In English, calculation involves numbers and the word usually connotes a simple process, but computation may be done by applying specific rules, with or without numbers, and the word is chosen for more complex ... Calculation is a prerequisite for computation ... The difference in the meaning of calculation and computation appears to originate from the late medieval period ...

Famous quotes containing the word calculation:

    “To my thinking” boomed the Professor, begging the question as usual, “the greatest triumph of the human mind was the calculation of Neptune from the observed vagaries of the orbit of Uranus.”
    “And yours,” said the P.B.
    Samuel Beckett (1906–1989)

    Common sense is the measure of the possible; it is composed of experience and prevision; it is calculation appled to life.
    Henri-Frédéric Amiel (1821–1881)