“Laurent Series”的概念、定义、习题解题方法及翻译-数学辞典

单词

Laurent Series

释义

Laurent Series

Let there be two circular contours and , with the radius of larger than that of . Let beinterior to and , and be between and . Now create a cut line between and ,and integrate around the path , so that the plus and minus contributions of cancel oneanother, as illustrated above. From the Cauchy Integral Formula,




			(1)

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



			(2)

For the first integral,

. For the second,

. Now use the Taylor Expansion (valid for

)

(3)

to obtain





			(4)

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

(5)

Now, use the Cauchy Integral Theorem, which requires that any Contour Integral of a function which enclosesno Poles has value 0. But

is never singular inside

for

, and

is never singular inside

for

. Similarly, there are no Poles in theclosed cut

. We can therefore replace

and

in the above integrals by

without altering theirvalues, so




			(6)

The only requirement on is that it encloses , so we are free to choose any contour that does so. The Residues are therefore defined by

(7)

See also Maclaurin Series, Residue (Complex Analysis), Taylor Series

References

Arfken, G. ``Laurent Expansion.'' §6.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 376-384, 1985.

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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