Articles of complex analysis

Convergence of Roots for an analytic function

Show that the roots of $$ f(z) = z^n+z^3+z+2 =0 $$ converge to the circle $|z|=1$ as $n \to \infty$.

Prove that $|\sin z| \geq |\sin x|$ and $|\cos z| \geq |\cos x|$

For any $z=x+iy$, prove the following: $$|\sin z| \geq |\sin x|$$ $$|\cos z| \geq |\cos x|$$ $\epsilon$-$\delta $ proof is not required. I don’t really know how to proceed. I know in order to remove the absolute values I can square both sides and I have tried proving this statement using the hyperbolic forms and […]

Is there a Möbius transformation that scales disks to the unit disk?

When working in the complex plane, often times I would like to scale a disk $|z-z_0|<R$ to the unit disk. I would first translate $z_0$ to the origin, but after that, what can we multiply by to scale the radius down from $R$ to $1$? I’m curious because in reading a proof of Schwarz’ lemma, […]

Contour Integration – my solution for real integral is complex?

So I’ve had a crack at this contour integration question and have somehow managed to get a complex solution for a real integral… I’ve gone through my working a number of times but can’t seem to find the mistake, so was hoping someone here could help. Evaluate the integral $$ I=\int^\pi_{-\pi} \frac{\,d\theta}{a+b\cos\theta+c\sin\theta} $$ where $a$, […]

Is this function holomorphic at 0?

This is for homework in my complex analysis class, and I think there may be a mistake. I wanted to make sure I didn’t miss anything obvious before I bring it up to the professor. The problem asks to show that the function $$ f(z) = \begin{cases} e^{-\frac{1}{z^4}}, & \text{if } z \neq 0 \\ […]

I cannot see why Ahlfors' statement is true (Extending a conformal map)

In page 234 in Ahlfors’ complex analysis text, the author talks about extending a conformal map. During the proof he states: We note further that $f'(z) \neq 0$ on $\gamma$· Indeed, $f'(x_0)= 0$ would imply that $f(x_0)$ were a multiple value, in which case the two subarcs of $\gamma$ that meet at $x_0$ would be […]

Determine the number of zeros of the polynomial $f(z)=z^{3}-2z-3$ in the region $A= \{ z : \Re(z) > 0, |\Im(z)| < \Re(z) \}$

Question: a). Determine the number of zeros of the polynomial $$f(z)=z^{3}-2z-3$$ in the region $$A= \{ z : Re(z) > 0, |Im(z)| < Re(z) \}$$. (b). Find the number of zeros of the function $$g(z) = z^3-2z-3+e^{-z^{2}}$$ in the region A. Comments: I think this is a fairly difficult problem. I assume that you have […]

Power of a function is analytic

This question already has an answer here: Show that if $f(z)$ is a continuous function on a domain $D$ such that $f(z)^N$ is analytic for some integer $N$, then $f(z)$ is analytic on $D$. 1 answer

Branches of analytic functions

Find a branch of $\log(z^2+1)$ that is analytic at $z=0$ and takes the value of $2\pi i$ there. Also, determine a branch of $\log(z^2+2z+3)$ that is analytic at $z=-1$. If I plug in $z=0$ and $z=-1$ to there respective functions then I get $\log(1)$ and $\log(2)$ but then what do I have to do to […]

Singular points

Find all the (isolated) singular points of $f$, classify them, and find the residue of $f$ at each singular point. $$f(z) = \frac{z^{1/2}}{z^2 + 1}$$ I think I have $3$ singularities at $z=0,-i$ and $i$ but am unsure about what to do next and what type of singularities they are. I don’t fully understand singularities […]