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 229 · Answer 1 · 2018-11-12 16:22:10

229

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 12 Nov 2018 - 16:22

score 806 · Answer 2 · 2026-03-24 02:34:37

806

Views

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

Published on 24 Mar 2026 - 2:34

score 63 · Answer 3 · 2018-11-16 08:08:02

63

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 16 Nov 2018 - 8:08

score 28 · Answer 4 · 2018-11-19 11:14:28

28

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 19 Nov 2018 - 11:14

score 188 · Answer 5 · 2018-11-27 15:03:28

188

Views

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

Published on 27 Nov 2018 - 15:03

score 49 · Answer 6 · 2018-12-03 18:29:27

49

Views

Analytic properties of a series

Published on 03 Dec 2018 - 18:29

score 150 · Answer 7 · 2018-12-03 22:43:45

150

Views

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

Published on 03 Dec 2018 - 22:43

score 139 · Answer 8 · 2018-12-10 21:14:00

139

Views

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

Published on 10 Dec 2018 - 21:14

score 434 · Answer 9 · 2026-02-23 04:48:51

434

Views

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

Published on 23 Feb 2026 - 4:48

score 94 · Answer 10 · 2018-12-12 04:01:20

94

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 12 Dec 2018 - 4:01

score 1k · Answer 11 · 2018-12-15 14:00:23

1k

Views

Definition of analytic function at a point

Published on 15 Dec 2018 - 14:00

score 414 · Answer 12 · 2018-12-29 12:52:45

414

Views

Fourier Transform of Exponentially Decaying Function Cannot Have Compact Support

Published on 29 Dec 2018 - 12:52

score 1.4k · Answer 13 · 2019-01-02 20:07:46

1.4k

Views

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

Published on 02 Jan 2019 - 20:07

score 106 · Answer 14 · 2019-01-08 01:58:54

106

Views

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

Published on 08 Jan 2019 - 1:58

score 168 · Answer 15 · 2026-03-24 04:55:11

168

Views

Solve equations of mixed type analytically

Published on 24 Mar 2026 - 4:55

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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