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Principal Part -- from Wolfram MathWorld

  • ️Weisstein, Eric W.
  • ️Tue Jun 20 2000

If a function f has a pole at z_0, then the negative power part

sum_(j=-k)^(-1)a_j(z-z_0)^j

(1)

of the Laurent series of f about z_0

sum_(j=-k)^inftya_j(z-z_0)^j

(2)

is called the principal part of f at z_0. For example, the principal part of

(z^2+1)/(sin(z^3))=z^(-3)+z^(-2)+1/6z^3+1/6z^4+...

(3)

is z^(-3)+z^(-2) (Krantz 1999, pp. 46-47).

See also

Laurent Polynomial, Laurent Series

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References

Krantz, S. G. "Principal Part of a Function." §4.3.1 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 46-48, 1999.

Referenced on Wolfram|Alpha

Principal Part

Cite this as:

Weisstein, Eric W. "Principal Part." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PrincipalPart.html

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