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Laurent Series

If f(z) isanalytic 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

			(2)
			(3)

(Korn and Korn 1968, pp. 197-198).

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 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 theCauchy integral formula,

			(4)
			(5)
			(6)

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

			(7)
			(8)
			(9)

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

(10)

to obtain

			(11)
			(12)
			(13)

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 of radius with r_1<=r<=r_2 , then

			(15)
			(16)
			(17)

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.

Thecomplex 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 aTaylor series with coefficients a_k .

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References

Arfken, G. "Laurent Expansion." §6.5 inMathematical 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 inTheory 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 inHandbook 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 inMethods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 374-398, 1953.

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Laurent Series

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Goodmanson, David andWeisstein, Eric W. "Laurent Series." FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/LaurentSeries.html

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Laurent Series

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