Complex Integral Evaluation for f(z)=z^4 on |z|=2 using Theorems 1 and 2

[John O' Meara]

Evaluate $$\int_C f(z)dz$$ by theorem 1 and check your result by theorem 2 where f(z) = z^4 and C is the semicircle |z|=2 from -2i to 2i in the right half-plane.
Theorem 1 : $$\int_{z_0} ^{z_1} f(z)dz = F(z_1) - F(z_0) \ \ \frac{dF}{dz}=f(z) \\$$
$$\int_{-2\iota} ^{2\iota}z^4dz = \frac{z^5}{5} = \frac{2^6 \iota}{5} \\$$
Theorem 2: $$\int_C f(z)dz = \int_a ^bf(z(t)) \frac{dz}{dt}dt\\$$. The path C can be represented by $$z(t)=2\exp{\iota t} \ f(z)=(2\exp{\iota t})^4 \ = \ 2^4\exp{4\iota t}\ \mbox{ therefore }\ \frac{dz}{dt}= 2 \iota\exp{\iota t} \\$$ $$\ \mbox{ therefore, the integral}\ = 2^5\iota\int_{\frac{-\pi}{2}} ^{\frac{\pi}{2}} \exp{5\iota t}dt \ = \ 2^5\iota\frac{\exp{6it}}{6\iota} \ = \ \frac{2^5}{3}(\frac{\exp{3\pi \iota} - \exp{-3\pi \iota} }{2}) =0 \\$$. Both theorems should give the same answer, but I don't get the same answer. Where have I gone wrong. Help please. Thanks.

 

[Dick]

In the last line, how did exp(5it) become exp(6it) after the integration?

 

[John O' Meara]

What is it then?

 

[Dick]

What's the integral of exp(k*x)dx? I don't think it has a k+1 in it.

 

[John O' Meara]

Thanks.