Compute $\oint_\gamma\sin z\, dz$

Question

Compute $$\oint_\gamma\sin z\, dz$$ where $\gamma =z_0 +e^{i\theta}$

My attempt : I was thinking about residue theorem , but i don't know how to apply residue theorem and compute

Answer 1

The integral of any entire function along any closed path is $0$. In the specific case of the sine function, you use the fact that you know a primitive of it (which is $-\cos$) to reach the same conclusion.

Answer 2

Note that the given function is holomorphic in the whole complex plane, and so it has no singularities at all. If you then look at the statement of the residue theorem, you see that the terms determining the value of the integral are only the values of the residue of the function at its singularities, which are none. So, in this case, the integral has value zero.

Another way to look at it is the following: When speaking of holomorphic functions (for example, $f \colon D \subset \mathbb{C} \longrightarrow \mathbb{C}$), if you can find a primitive of the function (that is, a holomorphic function $F$ with $F' = f$ in $D$), then the integral of $f$ along any path $\gamma \colon [0,1] \longrightarrow D$ with $\gamma (0) = a $ and $\gamma (1) = b$ will be exactly $F(b) - F(a)$ (In this case, since it is a closed path, it would then be zero). The implication I just referred to is actually an equivalence.