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Let $f\in \operatorname{Hol}(\mathbb{C} \setminus \{1\})$ and assume that $\int_{|z|=4}f(\xi) \, d\xi=0$.

Proe that for every closed and continuous curve $\gamma$ (closed - $\gamma(a)=\gamma(b)$) :

$$\int_\gamma f(z) \, dz=0.$$

I have no idea how to do that... any ideas?

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    Show that any such integral reduces to $n(\gamma)\int_{|z|=4} dz\;f(z)$ for some integer $n(\gamma)$. – 2017-01-03
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    What's the residue of $f$? Now apply the residue theorem to integrate $\gamma$. – 2017-01-03
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    You want $\gamma$ not to be just continuous, but a contour. Also you want the contour to lie in $\mathbb C \setminus \{1\}.$ – 2017-01-03
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    One shouldn't keep alternating in and out of MathJax in an expression written in mathematical notation. I've edited the question to put $f\in \operatorname{Hol}(\mathbb C \setminus \{1\})$ within just one pair of dollar signs. – 2017-01-03

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A theorem says that if $f$ is holomorphic in simply connected open set and $\gamma$ is a closed curve in that set, then $\displaystyle \int_\gamma f(z)\,dz=0.$ That takes care of the case where $\gamma$ does not wind around the number $1$.

Now suppose it winds once around $1$ in the same direction in which $|z|=4$ winds around that point -- they both run counterclockwise. Then look at two curves: Let $\alpha$ be a curve that includes some arc $\gamma_1$ of $\gamma,$ then goes along a straight path $\delta_1$ (not passing through $1$) from the end point of $\gamma_1$ to a point on $|z|=4,$ then goes halfway around the circle $|z|=4$ in the clockwise direction, then returns to the starting point of $\gamma_1$ along a straight path $\delta_2$. Let $\beta$ be a curve that follows the complementary arc $\gamma_2$ of $\gamma$, running from the starting point of $\delta_1$ along $\gamma$ to the end point of $\delta_2$, then follows $-\delta_2$ (i.e. $\delta_2$ in the direction opposite the direction that is included in the path $\alpha$), then continues along the circle $|z|=4$ in the clockwise direction to the end point of $\delta_1$, then follows $-\delta_1$ back to the starting point of $\gamma_2$.

Draw the picture and see if you can figure out why $$ \int_\alpha f(z)\,dz=0 \text{ and } \int_\beta f(z)\,dz = 0. $$ Then try to figure out why that, plus what is given, implies $\displaystyle\int_\gamma f(z)\,dz=0.$

Then try to figure out what to do with other curves, whose winding number around $1$ is something other than $0$ or $1.$

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    Does it have anything to do with residues? How can I draw conclusions about closed paths that are not $|z|=4$? Should I simply assume this is an indication for the residue? – 2017-01-04
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    @Meitar : You don't need explicit knowledge of residues to understand this answer. But you do need to understand the picture that the answer describes. – 2017-01-04