Compute the integral $$\int_{C}\cos\left(\frac{z}{2}\right)\,dz$$ Where
$C : γ(t) := ${$t + i\sqrt{π^2 − t^2}, −π ≤ t ≤ π$}
Not quite sure about how to go about doing this one, usually I parameterised and it became easy.
Complex integration problem?
1
$\begingroup$
integration
definite-integrals
parametric
complex-integration
1 Answers
2
The cosine function is an entire function.
Hence, the value of the integral depends only on the end points of the curve $C$. When $t=-\pi$, $z=-\pi$ and when $t=\pi$, $z=\pi$.
Therefore, we can write
$$\int_C \cos(z/2)\,dz=\left.\left(2\sin(z/2)\right)\right|_{z=-\pi}^{z=\pi}=4$$