All Questions
35
questions
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33
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Maximizing the radius of convergence around a point for an analytic function
Let $f:\Omega\longrightarrow \mathbb{C}$ be analytic, and let $z_0\in\Omega$ s.t.
$$f (z)=\sum_{n\geq 0} a_n(z-z_0)^n,\quad\forall \left|z-z_0\right|<r,$$
for some $r>0$, and some complex-valued ...
- 67
0
votes
0
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325
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Complex Analysis and Bernoulli Numbers from $\frac{z}{2} \cot (\frac{z}{2})$
Define the Bernoulli numbers $B_n$ by $\frac{z}{2} \cot (z/2) = 1 - B_1 \frac{z^2}{2!} - B_2 \frac{z^4}{4!} - B_{3} \frac{z^6}{6!} - ...$ Explain why there are no odd terms in this series. What is the ...
- 8,040
4
votes
1
answer
1k
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An analytic continuation of the square root along the unit circle
I want to find an analytic continuation of the square root along the unit circle but I am not sure whether I am doing it correctly.
Let $C_0$ be the open disk of radius $1$ around $1$, and let $f_0:...
- 1,237
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votes
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45
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Doubt regarding proving analyticity of a power series
I am self studying concepts of complex analysis from Ponnusamy and Silvermann "Complex Variables with Applications"
In the chapter of analytic continuation in basic concepts authors mention ...
user775699
2
votes
1
answer
61
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How to justify this statement?
Let $f$ a holomorphic function in $\mathbb{D}(0,1)$ such that $f(z)=\sum_{n\ge 0}a_n z^n$. Then apparentely we can deduce that for all $r\in [0,1[$, $n\ge1$ : $\int \limits_{0}^{2\pi} \overline{f(r\...
- 3,330
2
votes
1
answer
455
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An even function, $f(z)$, analytic near 0 can be written as another analytic function, $h(z^2)$
If f is an even function, $f(z) = f(-z)$, and is analytic near 0, then there exists a function h, also analytic near 0, such that $f(z) = h(z^2)$
I suspect this statement is true because the ...
- 475
1
vote
2
answers
235
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extending convergence radius using one boundary point
Show that if $f(z)=\sum\limits_{n\geq0}a_nz^n$ is analytic in $\{z\in\mathbb{C}:|z|<1\}\cup\{1\}$ and $\forall n\geq0:a_n\geq0$ then the radius of convergence of the power series is strictly larger ...
- 2,495
0
votes
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27
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Rate of convergence of the series for complex function
Suppose $f(z)=\sum a_n(z-z_0)^n$ and when $|z-z_0|\leq r<R$, there is a constant C so that $|f(z)-f(z_0)|\leq C|z-z_0|$. Suppose $|z-z_0|\leq r<R$, I try to show $\left|f(z)-\sum_{m=0}^{k}a_m(z-...
- 477
2
votes
2
answers
1k
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$\arctan(z)=\sum\limits_{n=0}^\infty \frac{(-1)^n}{2n+1} z^{2n+1}$ from $(\arctan(z))'=\frac{1}{1+z^2}=\sum\limits_{n=0}^\infty (-1)^n z^{2n}$
I'm at Kreyszig - "Advanced Engineering Mathematics" 10th ed. - sec. "15. Power Series, Taylor Series" - example 6. It finds the Maclaurin series of $f(z)=\arctan(z)$ by integrating
\begin{align*}
f'(...
- 311
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votes
1
answer
156
views
let $g(z)=f(\overline z)$ ,is $g$ analytic in $\Omega$ ??, [closed]
let $f$ is analytic in domain $\Omega$ in complex plane $\mathbb{C}$
let $g(z)=f(\overline z) \ \forall \ z \in \ \Omega $
is $g$ analytic in $\Omega$ ??
if not why??
i was trying it by C-R ...
- 2,671
1
vote
1
answer
50
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how to find this analytic function satisfying such condition
Decide whether there exists analytic $ f$ in $\mathbb{C}$ such that $f(n) = \cos(\sqrt{n})$ for all $n \in \mathbb{N}$.
I tried to raise this issue by Taylor's expansion, but I could not find a ...
- 13
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votes
5
answers
78
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If $g(x)=\sum_{n} f_n x^n=\frac{2x^2+x^3}{1-x-x^2}$,then find the general expression for the coefficients,$f_n$.
As evident $f_n=\frac{1}{n!}\frac{d^n}{dx^n}g(x)(at x=0)$.If I use Cauchy's integral formula to find the $nth$ derivative,then I'm stuck,because there also the derivative crops up while finding the ...
- 73
2
votes
2
answers
202
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Singularities of quotient of polynomials where the degree of the denominator $\ge$ the degree of the numerator $+2$.
Let the degrees of the polynomials
$$P(z)=a_0+a_1 z+a_2 z^2+\cdots +a_n z^n \; (a_n \neq 0)$$
and
$$Q(z)=b_0+b_1 z+b_2 z^2+\cdots +b_m z^m \; (b_m\neq 0)$$
be such that $m \ge n+2.$ Show that if ...
- 5,601
0
votes
0
answers
122
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Find an analytic continuation
Let $f(z)=\sum_{j=0}^{\infty}z^j$ for $|z|<1$. For what values of $\alpha$ ($|\alpha|<1$) does the Taylor expansion of f(z) about $z=\alpha$ yield a direct analytic continuaton of f(z) to a disk ...
- 455
3
votes
1
answer
164
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If $f,g$ real analytic and $\lim_{t \to t_0} f(t)/g(t)$ exists then $f/g$ is analytic
If $f,g$ are real analytic at $t_0$ and $\lim_{t \to t_0} f(t)/g(t)$ exists then is it true that $f/g$ with the limiting value filled in at $t= t_0$ is real analytic at $t_0$? I know the complex ...
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