AMAT 415 — Week 8 problems

40) Find the first 5 terms of the Taylor series of f(z)=e^z\sin(z) around z=0.

41) Find the Taylor series of f(z)=\sin(z+\pi/4) around z=0. (Suggestion: Use the identity \sin(x+y) = \sin(x)\cos(y) + \cos(x)\sin(y).)

42) Find the first four nonzero terms in the Taylor series of

\displaystyle{f(x)=\frac{(\cos(x)-1)^2}{x^4}}

around z=0.

43) Find the Taylor expansion of

\displaystyle{f(x)=\frac{1}{(z-1)^2(z-2)}}

around z=0. (Suggestion: First do a partial fraction decomposition.)

44) Find the Taylor expansion of

\displaystyle{f(x)=\frac{1}{(z+i)^2}}

around z=i.

45) Find the first 5 nonzero terms in the Laurent expansion of 1/\sin(z) around z=0.

46) Find the terms of nonpositive index in the Laurent expansion of

\displaystyle{f(x)=\frac{e^z+1}{(z^2-1)^2}}

around z=1.

47) Find the points where

\displaystyle{f(x)=\frac{1}{z^3(z+4)}}

is singular (i.e. not analytic). Find its residues at those points.

48) Same as the previous question, but with

\displaystyle{f(x)=\frac{1}{z^3-3}}.

49) Same as the previous question, but with

\displaystyle{f(x)=\frac{e^z}{z(1-z)^3}}.

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