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Questions tagged [holomorphic-functions]

For questions on holomorphic functions, complex-valued functions of one or more complex variables that are complex differentiable in a neighborhood of every point in its domain.

A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain. The existence of a complex derivative in a neighborhood is a very strong condition, for it implies that any holomorphic function is actually infinitely differentiable and equal to its own Taylor series (analytic). Holomorphic functions are the central objects of study in complex analysis.

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About holomorphic function which avoids $\mathbb R$

I was asking myself whether or not the following statement is true, and I think it is. The conjecture Let $f$ be an entire function, i.e. $f$ holomorphic and $f:\mathbb C\to \mathbb C$. We assume that $$f(\mathbb C)\cap \mathbb R=\emptyset.$$ Then…
E. Joseph
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An application of Weierstrass theorem for holomorphic functions

We have the following theorem for holomorphic functions. Theorem (Weierstrass) If $(f_n)$ is a sequence of holomorphic functions on an open set $U\subset\mathbb C$ such that $(f_n)$ tends uniformly to $f$ on every compact $K\Subset U$. Then $f$ is…
E. Joseph
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bijective holomorph function

I have a holomorphic function $f: G \rightarrow \mathbb{C} $. G is a domain and $ G \subset \mathbb{C} $. Furthermore $ V, V^{*}$ are domains, too, with continous differentiable closed boundary curves $ \Gamma, \Gamma^{*}$ and $\bar{V} \subset…
Leon1998
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Harmonic holomorphic function in $Ω$

$Ω$ is simply connected in $C$, $u$ is a harmonic function in $Ω$ , $v$ in $Ω$ $$v(x,y) = \int_0^1 (yu{\Tiny x} (sx,sy)-xu{\Tiny y} (sx,sy)) ds$$ Prove that there exists a holomorphic function $u+iv$ in $Ω$ I know the Cauchy–Riemann equations and…
malilini
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Is modulus of a complex function always non holomorphic?

Let $f(z)$ be a complex function that is holomorphic on an open subset of the complex plane. Now, if we define another function $g(z)=|f(z)|^2$, can we say anything about the holomorphicity of $g(z)$ for arbitrary $f(z)$? Also, can you give examples…
damaihati
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Holomorphic domain of $ \Phi\left(z\right)=\ln\left(\frac{z-a}{z+a}\right)$

I have to find a domain $D$ of the complex plane where the function ( $a \in \mathbb{R}^{*+}$ ) defined by $$ \Phi\left(z\right)=\ln\left(\frac{z-a}{z+a}\right) $$ is holomorphic. Hence i search for the set of points where it is not holomorphic. I…
Atmos
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Change of variable in integral for $f(z,\bar{z})$

Consider the integral of a function of two variables : $$I=\int dx dy f(x,y)$$ Now we do the change of variables : $z=\frac{x+iy}{\sqrt{2}}$, $\bar{z}=\frac{x-iy}{\sqrt{2}}$. In my course it is written that we have then : $$I=-i\int dz d\bar{z}…
StarBucK
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Holomorphic on the ball continuous up to the boundary can have a non isolated zero on the boundary?

Let $U = B(0,1)\subset \mathbb{C}$ be the unit ball, and $f\in A(U)= H(U)\cap C^0(\overline{U})$ non constant where $H(\Omega)$ are the holomorphic functions over $\Omega$. We know that if $z_0 \in U$ is a zero of $f$ than it must be isolated…
Overflowian
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Is a function with surface like a bottom of a boat holomorphic.

Consider a surface somewhat like bottom of a boat. Imagine cutting a paraboloid (see picture) in half, pulling the halves at a distance and joining the two halves with a matching parabolic cylinder (see picture). In other words consider $f(z)$ for…
Shree
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Why is a function like $z^{2.5}$ not holomorph?

If I consider the function $f(z)=z^{2.5}$ $\in \mathbb{R}$, it is continuously differentiable with $f'(z)=2.5z^{1.5}$. But why can I not do the same in $\mathbb{C}$ with $ f'(z)=2.5z^{1.5}$?
tim123
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Harmonic holomorphic function in Ω

$Ω$ is simply connected in $C$, $u$ is a harmonic function in $Ω$ , $v$ in $Ω$ $$v(x,y) = \int_0^1 (yu{\Tiny x} (tx,ty)-xu{\Tiny y} (tx,ty)) dt$$ Prove that there exists a holomorphic function $u+iv$ in $Ω$ I have the solution - can someone…
malilini
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If the derivatives of $f$ for $t\in L$ are real functions, then $f(\bar{z})=\overline{f(z)}, \forall z\in B(t,r).$

Let $L\subset \mathbb{R}$ be an open interval and $f$ be a holomorphic function on $B(t,r)$ where $t\in L.$ How to prove the claim: If the derivatives of $f$ for $t\in L$ are real functions, then $$f(\bar{z})=\overline{f(z)}, \forall z\in…
Emo
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Let $f:\bar{U}\rightarrow\mathbb{C}$ a continuous function such that $f|_U$ is holomorph. Find $f$ where $f(t)=\cos t+e^t$ if $t\in[0,1]$

$U$ is the square with verices in $0, 1, i, i+1$. I though immediately using the identity principle. The thing is what I know about it is that, if U is an open and connected set, holomorph in U such that exists $z_n, f, g$ such that $f(z_n)=g(z_n)$…
Silkking
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