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Laurent Series -- from Wolfram MathWorld

  • ️Weisstein, Eric W.
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If f(z) is analytic throughout the annular region between and on the concentric circles K_1 and K_2 centered at z=a and of radii r_1 and r_2<r_1 respectively, then there exists a unique series expansion in terms of positive and negative powers of (z-a),

f(z)=sum_(k=0)^inftya_k(z-a)^k+sum_(k=1)^inftyb_k(z-a)^(-k),

(1)

where

(Korn and Korn 1968, pp. 197-198).

LaurentSeries

Let there be two circular contours C_2 and C_1, with the radius of C_1 larger than that of C_2. Let z_0 be at the center of C_1 and C_2, and z be between C_1 and C_2. Now create a cut line C_c between C_1 and C_2, and integrate around the path C=C_1+C_c-C_2-C_c, so that the plus and minus contributions of C_c cancel one another, as illustrated above. From the Cauchy integral formula,

Now, since contributions from the cut line in opposite directions cancel out,

For the first integral, |z^'-z_0|>|z-z_0|. For the second, |z^'-z_0|<|z-z_0|. Now use the Taylor series (valid for |t|<1)

1/(1-t)=sum_(n=0)^inftyt^n

(10)

to obtain

where the second term has been re-indexed. Re-indexing again,

f(z)=1/(2pii)sum_(n=0)^infty(z-z_0)^nint_(C_1)(f(z^'))/((z^'-z_0)^(n+1))dz^' +1/(2pii)sum_(n=-infty)^(-1)(z-z_0)^nint_(C_2)(f(z^'))/((z^'-z_0)^(n+1))dz^'.

(14)

Since the integrands, including the function f(z), are analytic in the annular region defined by C_1 and C_2, the integrals are independent of the path of integration in that region. If we replace paths of integration C_1 and C_2 by a circle C of radius r with r_1<=r<=r_2, then

Generally, the path of integration can be any path gamma that lies in the annular region and encircles z_0 once in the positive (counterclockwise) direction.

The complex residues a_n are therefore defined by

a_n=1/(2pii)int_gamma(f(z^'))/((z^'-z_0)^(n+1))dz^'.

(18)

Note that the annular region itself can be expanded by increasing r_1 and decreasing r_2 until singularities of f(z) that lie just outside C_1 or just inside C_2 are reached. If f(z) has no singularities inside C_2, then all the b_k terms in (◇) equal zero and the Laurent series of (◇) reduces to a Taylor series with coefficients a_k.

See also

Complex Residue, Maclaurin Series, Principal Part, Taylor Series

Portions of this entry contributed by David Goodmanson

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References

Arfken, G. "Laurent Expansion." §6.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 376-384, 1985.Korn, G. A. and Korn, T. M. Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill, p. 198, 1968.Knopp, K. "The Laurent Expansion." Ch. 10 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, pp. 117-122, 1996.Krantz, S. G. "Laurent Series." §4.2.1 in Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 43, 1999.Morse, P. M. and Feshbach, H. "Derivatives of Analytic Functions, Taylor and Laurent Series." §4.3 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 374-398, 1953.

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Laurent Series

Cite this as:

Goodmanson, David and Weisstein, Eric W. "Laurent Series." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LaurentSeries.html

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