Questions tagged [analyticity]

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 of functions over the complex numbers.

If $U$ is a subset of $\mathbb{C}$, $f$ is analytic at $x_0$ if there exists a series $$ \sum_{j=0}^\infty a_j (z-z_0)^j $$ that converges to $f$ at a neighbourhood of $z_0$. In complex analysis, analyticity is equivalent to homomorphy.

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Finding all the analyitical function in the unit annulus that satisfy a given condition for natural numbers

Let $f$ be an analytic function in the annulus $0 < |z| < 1 $ such that it's singularity in $z=0$ is not essential. I want to find all of such functions $f$ that satisfy for $n = 3, 4,...$: $f(\frac 1n)=\frac {n^4}{1+n}$ $f(\frac 1n)=\frac…
Snufsan
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Sequence of partial sums converges locally uniformly?

Suppose $f: U \to \mathbb{R}$ is a real analytic function defined by $f(x)=\sum_1^\infty a_n (x-x_0)^n$ and let $f_N=\sum_1^N a_n(x-x_0)^n$. Then $\{f_N\}$ converges to $f$ pointwise. Wikipedia says this convergence must be locally uniform, but…
goatman2743
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how to prove that a function is not complex differentiable

I was working on a problem on the complex differentiability of the following function: $f(z)= z \operatorname{Re}(z)$. How to find the points where the given function is not differentiable. My effort: $f(z)= z\operatorname{Re}(z) = zx$ where $x$ is…
monalisa
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Real valued and holomorphic function

I was wondering about this problem for a while out of curiosity: is there a non-constant analytic function with real values on $\mathbb{R}$ and purely imaginary values $i\mathbb{R}$? I think the answer is a no by using Cauchy-Riemann equations…
Spock
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Proving that $f(z)\neq \frac{z}{z+1}$ in $D_1(0)$

Suppose $f$ is analytic in $D_r(0)$ for some $r>1$. I want to prove that $f(z)\neq \frac{z}{z+1}$ in $D_1(0)$. This is how I tried to prove this. Assume $f(z)= \frac{z}{z+1}$ in $D_1(0)$. Now define $a_n=-1+\frac{1}{n},n\in \mathbb{N}$. Then $a_n…
Heisenberg
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A function property to guarantee that being constant on an interval implies identically constant

Let $f:\mathbb R\rightarrow \mathbb R$. Suppose we know that $f $ is a constant on some open/closed interval. Which condition does guarantee that $f $ is constant on $\mathbb R$? Clearly, continuity is not enough. Differentiable? $C^1$? smooth? real…
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Function holomorphic in the neighb. of zero, bounded by exponent is equal 0

I want to prove that if $f$ is a holomorphic function in a neighbourhood of $0$ and $|f(\frac{1}{n})| \le \frac{1}{e^n}$ for $n$ sufficiently big, then $f =0$. I know that if $f$ is holomorphic in a neighb. of zero, then it has the form…
Spencer
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Can you always specify the value of a $C^{\infty}$ or analytic function on an isolated set?

Former math grad student, now a lawyer for the last $28$ years. Just doing math for fun in my spare time. I was browsing questions here and my mind went on a tangent. The following questions occurred to me, which I strongly suspect are…
Robert Shore
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Real Analytic continuation

For which values of $p,q\in[1,\infty)$ the following functions have a real analytic continuation to the whole real line. 1.$f:(0,\infty)\to \mathbb R$ where $f(t)=(1+t^p)^{\frac{q}{p}}$. 2.$g:(-\infty,0)\to \mathbb R$ where…
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Zero set of a non constant analytic function.

Is there any example of a non constant analytic function on { z : |z|<1} , which have infinite zeros in that domain?
A learner
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If $u=ax^3+by^3$ and $u$ is harmonic, find values of $a$ and $b$. Also find the harmonic conjugate of $u$.

If $u=ax^3+by^3$ and $u$ is harmonic, find values of $a$ and $b$. Also find the harmonic conjugate of $u$. I could not find any confirmation regarding this solution of $a$ and $b$.
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Analytical functions equal at a point implies all derivatives are also equal at that point?

There is this problem that goes as follows: If I have two analytical functions $f,g:I→R$ (where $I$ is an open interval), and I know that there is a point $a∈I$ where $f(a)=g(a)$ and also $f^{(k)}(a)=g^{(k)}(a), \enspace \forall k \in \mathbb{R}$…
H44S
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How to prove the following using residue theorem?

I have this HW and I don't know how to approach it, does anybody know how can it be shown that $$∑_{n=1}^∞\frac{1}{n^2} =\frac{π^2}{6}$$ using residue theorem?
yasiren
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Find the analytic function $()=+$ such that $−=(+)(−)$

Find the analytic function $()=+$ such that $−=(+)(−)$. I got $u_x=2x+v_x$ and $v_y=u_y+2y$. To be analytic $u_x=v_y$ and $u_y=-v_x$. Iam stuck after the above step.Tried summing both the equation ,still not sure about it
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Testing the analyticity of $2\ln z$

Test whether the function $2\ln z$ is analytic. I have tried to test this function for analyticity by letting $z=x+iy$, but I have failed to separate the real and imaginary parts. Any help would be appreciated.
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