# Tagged Questions

A function is analytic if it has a converging power series expansion. Please also use one of (real-analysis) or (complex-analysis) to specify real or complex analyticity: though the definition is the same, the properties of functions analytic over real numbers is quite different from the properties ...

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### Prove that if $f$ and $g$ are analytic at $w$, then so is $fg$

Prove that if $f$ and $g$ are analytic at $w$, then so is $fg.$ My main attempt was using the Cauchy-Riemann equations on the product in this manner but this did not work out. My thinking: \begin{...
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### Intersection of zero sets of real analytic functions of two variables

The zero set of a real analytic function cannot contain an open set. If we have two distinct real analytic functions of two variables, can they intersect in more than at isolated points? Since the ...
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### curl-free, conservative vector fields in complex analysis

I just verified that for the conjugate of an analytic function $\bar{f}$=u-iv, this conjugate function is curl-free - the Cauchy-Riemann equations forces this to be the case. Then we can consider f=u-...
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### How to understand the Identity Theorem in complex analysis, from the point of view of power series expansions

The theorem states that if $f$ and $g$ are analytic functions and their values agree on an open set that is contained in a larger, connected domain, then $f$ must equal $g$ on the entire domain. (The ...
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### Application of Baire Category theorem

Suppose that $f$ is infinitely differentiable on $[a,b]$ and suppose that for any $a ≤ x ≤ b$ the Taylor series of $f$ has positive radius of convergence at $x$. Use the Baire Category Theorem to ...
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### Conjecture on zeros of analytic function

I have a conjecture that I can´t prove nor disprove, any help on doing so will be very grateful. Let $f: \{z: |z|<2\} \to \mathbb C$ be a non constant analytic function such that if $|z|=1$ ...
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### Show that f must be constant on C

This is a problem that I have been encountered after reading about analytic functions in complex analysis. Suppose $f(z) = f(x + iy)$ is analytic on $\mathbb{C}$. Let $u= \Re ~f$ and $v = \Im ~f$. ...
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### On applying Whitney's extension theorem to suitable closed sets

Whitney's extension theorem states that if $D \subset \mathbb{R}^n$ is closed and $f: D \to \mathbb{R}$ is $C^k$ in some sense to be specified below, then $f$ can be extended to $\mathbb{R}^n$ so that ...
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### Not able to understand a paragraph in John Conway's Complex analysis book.

On page 97 under the heading "Counting zeroes; the open mapping theorem" there is a second paragraph which goes like this: In section 3 it was shown that if an analytic function $f$ had a zero ...
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### Prove that $\,\displaystyle f(z) = \sum_{n\ge1}\frac{z^n}{n^2}$ is univalent in the disk $\,D\big(\frac23\big)$

I'm having some difficulty with this question: Prove that the function $\,\,\displaystyle f(z) = \sum_{n=1}^\infty\frac{z^n}{n^2}\,$ is univalent in the disk $D\big(\frac23\big)$. There is the ...
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### smooth vs. analytic in the definition of almost-complex manifolds

Let $A_{\infty}\hspace{-0.03 in}$ be a maximal $C^{\infty}\hspace{-0.02 in}$ atlas on $M\hspace{-0.03 in}$, and with that smooth structure on $M$, suppose $\: j : TM\to TM\:$ is a smooth function ...
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### Measure of inverse image of points by an analytic mapping

How can one prove the following statement: For any analytic mapping from a connected analytic manifold $M$ to an analytic manifold $N$, the inverse image of a point in $N$ is either the whole of $M$ ...
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### Schwarz Reflection Principle for the four quadrants in the plane and for two intersecting circles,

I'm looking at an old exam problem that shows a picture of what the function f does to the plane. On the upper right quadrant, there is a + sign, which indicates that f maps this quadrant one-to-one ...
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### Analytic continuation of ln(z) counterclockwise about the unit circle,

We write ln(z) as ln(1+z-1) = ln(1+(z-1)) to utilize the familiar expansion that is: (z-1) - (z-1)^2 / 2 + ... which converges for |z-1| < 1, i.e., we get convergence of ln(z) in an open Taylor ...
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### Analyticity of log f(z)

In a solution to a problem, I read that, if $f(z)$ is entire, $f(z)\neq0$ and the domain of definition of $f(z)$ is simply connected, then it is possible to choose a branch of log $f(z)$ that is ...
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### Application of Bernstein's theorem

There is a theorem due to Bernstein related to analytic functions : If $f : ]0,1[ \to \mathbb{R}$ is an absolutely monotonous function (that is a $\mathcal{C}^\infty$ function such that for ...
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### Question about inverse of CDF being a real analytic function

Let F: [0,a] -> [0,1] be a continuous, strictly increasing CDF. Assume also F admists a continuous, positive pdf f. Now define the inverse function h(x) as F(h(x))=x. Is h real analytic? If not, what ...
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### Analyticity of $\dfrac{1}{z}$ vs. $\dfrac{1}{z^2}$

I am learning complex analysis on my own. I am familiar with the theorems, and I am able to compute by hand and get correct results. But there is something that escapes me. What is the criteria for ...
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### A Question from complex variable [closed]

Show that an analytic function with constant modulus is itself a constant
Let $f: \mathbb D \to \mathbb D$ be analytic or holomorphic with $f(0)=\frac{1}{2}$ and $f(\frac{1}{2}) = 0$ where $\mathbb {D} = \{ z: |z| \leq 1\}$. Then find $|f^{'}(0)|$ and $|f^{'}(\frac{1}{2})|$....
### Find the residue of the function $g(z)=f(z^2)$ at a given point.
Let $f(z)$ be analytic in $0<|z|<R$. Find the residue of the function $g(z)=f(z^2)$ at $z_0=0$. I am looking for a solution to this problem. My thoughts: I know in order to find the residue ...