Answer to Question #125656 in Complex Analysis for Natakani

Question #125656

Let f(z) = sin z/z

and f(0) = 0. Explain why f is analytic at z = 0. Find the Maclaurian 

series for f(z) and g(z) = ∫ f(ξ)dξ from 0 to z

. Does there exist a function f with an 

isolated singularity at 0 and such that |f(z)|~ exp( 1/|z|) near z= 0? 



1
Expert's answer
2020-07-08T19:40:18-0400

We can write

sin(z)=z-z3/6+z5/120-...+(-1)kz2k+1/(2k+1)!+...

Then

f(z)=sin(z)/z=1-z2/6+z4/120-...+(-1)kz2k/(2k+1)!+...

is Maclaurian series.

z->0 => f(z)->1

f(0)=0

So f(z) is not continuous function at z = 0 and therefore f(z) is not analytic at z = 0.

But if f(0)=1 then we have

"f(z)=\\sum_{n\\ge0}a_nz^n"

where

"a_n = \\begin{cases}\n 0 &\\text{if } n=2k+1 \\\\\n (-1)^{k}\/(2k)! &\\text{if } n=2k\n\\end{cases}"

"(2k)!\\backsim (2k\/e)^{2k}(2\\pi2k)^{1\/2}"

"\\sqrt[2k]{(2k)!}\\backsim (2k\/e)(2\\pi2k)^{1\/{4k}}\\rarr infinity"

and then

"\\sqrt[n]{|a_n|}\\rarr0".

Therefore the series is covergent on the complex plane and function f(z) is analytic at z=0.

g(z)="\\int" f(y)dy=C+z-z3/18+z5/600-...+(-1)kz2k+1/((2k+1)(2k+1)!)+...="C+\\sum_{n\\ge0}b_nz^n"

where "\\sqrt[n]{|b_n|}\\rarr0".

Therefore the series is covergent on complex plane and function g(z) is analytic and

"\\int"z0f(y)dy=(C+y-y3/18+y5/600-...+(-1)ky2k+1/((2k+1)(2k+1)!)+...)|z0=

=z-z3/18+z5/600-...+(-1)kz2k+1/((2k+1)(2k+1)!)+...

No. There is no function f with an 

isolated singularity at 0 and such that |f(z)|~ exp( 1/|z|) near z= 0

Because in this cace

"\\int"|z|=rf(z)dz does not depend on r.

But for f(z) this integral depends on r and it is equal to 2"\\pi" re1/r








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