Question

Find a function $f$ analytic at $x_{0} = 0$ so that $f\left(\frac{1}{n}\right) = \frac{n}{n + 1}, n = 1, 2, \ldots$.

score 231 · Answer 1 · 2026-04-17 05:32:32

231

Views

Find a function $f$ analytic at $x_{0} = 0$ so that $f\left(\frac{1}{n}\right) = \frac{n}{n + 1}, n = 1, 2, \ldots$.

Published on 17 Apr 2026 - 5:32

score 807 · Answer 2 · 2026-03-25 01:29:07

807

Views

If $f$ is an entire function with $|\,f(z)|\le|\operatorname{Re}(z)|$, then $\,f\equiv 0$.

Published on 25 Mar 2026 - 1:29

score 66 · Answer 3 · 2026-04-15 03:58:49

66

Views

If $f$ is analytic on $\mathbb{R}$, is it necessary that $f = \sum_{n = 0}^{\infty} a_{n} x^{n}$ converges for all $x \in \mathbb{R}$?

Published on 15 Apr 2026 - 3:58

score 31 · Answer 4 · 2026-04-11 04:26:25

31

Views

Is a function still analytic if its infinite sequence of subsequent derivatives, at every point of its domain, grow faster than the factorial?

Published on 11 Apr 2026 - 4:26

score 191 · Answer 5 · 2026-04-16 01:51:11

191

Views

Show that $h(z)=\int_{0}^{1}\frac{t^2}{1-tz}dt$ is analytic in a nbh. of zero.

Published on 16 Apr 2026 - 1:51

score 53 · Answer 6 · 2026-04-08 19:06:38

53

Views

Analytic properties of a series

Published on 08 Apr 2026 - 19:06

score 155 · Answer 7 · 2026-05-05 20:15:03

155

Views

Bounding the extrema of polynomials from $\frac{d^n}{dx^n} \exp(-1/x)$

Published on 05 May 2026 - 20:15

score 142 · Answer 8 · 2026-04-15 13:33:29

142

Views

Can Analytic Functions on the Circle be Characterized by Sobolev Norms?

Published on 15 Apr 2026 - 13:33

score 437 · Answer 9 · 2026-04-11 16:05:46

437

Views

An even function, $f(z)$, analytic near 0 can be written as another analytic function, $h(z^2)$

Published on 11 Apr 2026 - 16:05

score 97 · Answer 10 · 2026-04-17 10:53:17

97

Views

If $f$ is analytic on $(a, b)$ containing at point $x_{0}$ with $f^{(n)}(x_{0}) = 0$ for $n \in \mathbb{N}$, prove $f(x) = 0$ for all $x$.

Published on 17 Apr 2026 - 10:53

score 1k · Answer 11 · 2026-04-13 04:39:33

1k

Views

Definition of analytic function at a point

Published on 13 Apr 2026 - 4:39

score 417 · Answer 12 · 2026-04-09 07:07:14

417

Views

Fourier Transform of Exponentially Decaying Function Cannot Have Compact Support

Published on 09 Apr 2026 - 7:07

score 1.4k · Answer 13 · 2026-04-15 14:19:18

1.4k

Views

Determine if $f(z) = \log(e^z+1)$ is analytic and where

Published on 15 Apr 2026 - 14:19

score 108 · Answer 14 · 2026-04-16 08:22:20

108

Views

$f^{(k)}(z_0)=0$ for all $k\geq0$ implies that $f=0$

Published on 16 Apr 2026 - 8:22

score 170 · Answer 15 · 2026-03-25 14:23:45

170

Views

Solve equations of mixed type analytically

Published on 25 Mar 2026 - 14:23

List Question

Find a function $f$ analytic at $x_{0} = 0$ so that $f\left(\frac{1}{n}\right) = \frac{n}{n + 1}, n = 1, 2, \ldots$.

If $f$ is an entire function with $|\,f(z)|\le|\operatorname{Re}(z)|$, then $\,f\equiv 0$.

If $f$ is analytic on $\mathbb{R}$, is it necessary that $f = \sum_{n = 0}^{\infty} a_{n} x^{n}$ converges for all $x \in \mathbb{R}$?

Is a function still analytic if its infinite sequence of subsequent derivatives, at every point of its domain, grow faster than the factorial?

Show that $h(z)=\int_{0}^{1}\frac{t^2}{1-tz}dt$ is analytic in a nbh. of zero.

Analytic properties of a series

Bounding the extrema of polynomials from $\frac{d^n}{dx^n} \exp(-1/x)$

Can Analytic Functions on the Circle be Characterized by Sobolev Norms?

An even function, $f(z)$, analytic near 0 can be written as another analytic function, $h(z^2)$

If $f$ is analytic on $(a, b)$ containing at point $x_{0}$ with $f^{(n)}(x_{0}) = 0$ for $n \in \mathbb{N}$, prove $f(x) = 0$ for all $x$.

Definition of analytic function at a point

Fourier Transform of Exponentially Decaying Function Cannot Have Compact Support

Determine if $f(z) = \log(e^z+1)$ is analytic and where

$f^{(k)}(z_0)=0$ for all $k\geq0$ implies that $f=0$

Solve equations of mixed type analytically

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