Calculate Complex Integrals with Series of Sin Z

[rbnphlp]

By parameterizing the curve (not by Cauchy's theorem) and using the series of sin z, nd
the value of
∫z^k sin(z)dz around a closed Contour C where C is the unit circle z=e^(iθ), for 0≤θ<2π

What do they mean by using series of sin z ? I mean if I expand it .. I get e^(iθ)- e^(3iθ)/3! ---
and so on ..Not sure what Iam meant to do with that since its an infinite series ..Do I say for small z sinz≈z , but not sure ..

thanks

 

[tiny-tim]

hi rbnphlp!

What do they mean by using series of sin z ? I mean if I expand it .. I get e^(iθ)- e^(3iθ)/3! ---
and so on ..

they mean sinz = z - z³/3! + z⁵/5! - …

(learn it … also cosz = 1 - z²/2! + z⁴/4! - … )

Not sure what Iam meant to do with that since its an infinite series ..

(i haven't tried it , but …)

my guess is that all the integrals are going to be 0 except for one …

if so, it won't matter that there's an infinite number of them!