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单词 RegularAtInfinity
释义

regular at infinity


When the function w of one complex variable is regular in the annulus

ϱ<|z|<,

it has a Laurent expansion

w(z)=n=-cnzn.(1)

If especially the coefficients c1,c2, vanish, then we have

w(z)=c0+c-1z+c-2z2+

Using the inversionPlanetmathPlanetmathz=1ζ,  we see that the function

w(1ζ)=c0+c-1ζ+c-2ζ2+

is regular in the disc  |ζ|<ϱ.  Accordingly we can define that the function w is regular at infinity also.

For example,  w(z):=1z  is regular at the point  z=  and  w()=0.  Similarly, e1z is regular at and has there the value 1.

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更新时间:2025/5/5 0:14:20