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MAT334--Lectures & Home Assignments / 2.5 Q19
« on: November 18, 2018, 04:01:10 AM »
Can anyone help me this question? Thanks!
Suppose that the Laurent series $\sum\nolimits_{-\infty}^{+\infty}a _n (z-z_0)^n $ converges for $ f<| z-z_0|<R $ and
$$
\sum \limits_{-\infty}^{+\infty} a_n (z-z_0)^n=0, 0<|z-z_0|<r.
$$
Show that $a_n=0,n=0, \pm 1, \pm 2,\cdots$, (Hint:Multiply the series by $(z-z_0)^{-m}$ and integrate around the circle $|z-z_0|=s$, $r<s<R$ with respect to $z$. The result must be zero, but it is also $a_{m-1}$.)
Suppose that the Laurent series $\sum\nolimits_{-\infty}^{+\infty}a _n (z-z_0)^n $ converges for $ f<| z-z_0|<R $ and
$$
\sum \limits_{-\infty}^{+\infty} a_n (z-z_0)^n=0, 0<|z-z_0|<r.
$$
Show that $a_n=0,n=0, \pm 1, \pm 2,\cdots$, (Hint:Multiply the series by $(z-z_0)^{-m}$ and integrate around the circle $|z-z_0|=s$, $r<s<R$ with respect to $z$. The result must be zero, but it is also $a_{m-1}$.)