The roots of a polynomial in the first quadrant

Question

Let $f(z)=z^5+1$ be a polynomial in $\mathbb{C}$. Using Argument principle show that $f$ has one root in the first quadrant.

$$ \int_C\dfrac{5z^4}{z^5-1}\operatorname{d}z. $$ But argument principle is applicable for closed contour $C$. Then how will I apply for the first quadrant?

Answer 1

Let $C$ be the curve from $b$ on the positive real axis to $ib$ along the curve $be^{i\theta}$, then down the imaginary axis to $0$ finally back along the real axis to $b$. Take $b$ large. Look at the image of the curve under $f$, the quarter circle part goes around the origin once, the imaginary part drops down along the $x=1$ line and then to $b^5+1$, thus it winds around the origin once.

Answer 2

If $z^5=-1$ then $|z|^5 = 1$ so $|z|=1.$ Thus the curve need not surround the whole first quadrant, but only some part of it that includes all points $z$ in the first quadrant at which $|z|=1.$