All Questions
14
questions
0
votes
1
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98
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if $P(x_{0})=0$, but $\lim_{ x \to x_{0} }(x-x_{0})\left( \frac{Q(x)}{P(x)} \right) = \text{finite}$, then function is analytic at $x=x_0$
I'm currently learning about Differential Equations and the textbook I'm using (by DiPrima) says the following:
For polynomials $P(x)$ and $Q(x)$ that has no common factors, if $P(x_{0})=0$, but $\...
- 1,686
2
votes
2
answers
641
views
How to find the radius of convergence of series solution of an ODE?
Recently, I watched a video https://www.youtube.com/watch?v=8lHAMZWDQHI which taught me how to find the radius of convergence of power series solution of an ODE. For example, if we want to find the ...
- 437
1
vote
0
answers
38
views
Generating sequence of $\cosh a\sqrt{z}$ (recurrence)
Let ($z\in\mathbb{C}$, $a\in\mathbb{C}\setminus\{0\}$)
$$w=\cosh a\sqrt{z}=\sum_{n\ge 0}\frac{a^{2n}z^n}{(2n)!}.$$
Then $w$ satisfies the ODE
$$4zw''+2w'-a^2w.$$
So if $w=\sum_{n\ge 0}a_n(z-1)^n$, ...
- 359
1
vote
2
answers
536
views
Weierstrass M Test and Analytic Functions
I have studied how to solve some PDEs using Fourier's Method. But, I'd like to know why don't we just use Taylor Series to solve these. After a little research, I found two interesting results: 1st) ...
- 516
0
votes
1
answer
34
views
Find the solutions to the $w''-z^2w=3z^2-z^4$ as Taylor series where $w(0)=0$ and $w'(0)=1$
We need to find the solutions of the
$w''-z^2w=3z^2-z^4$
where
$w(0)=0;w'(0)=1$
I wrote down the series that we can use to find the answer ($w$ as Taylor series):
$w=\sum_{n=0}^\infty C_nz^...
- 283
2
votes
2
answers
387
views
Find analytic Functions such that $f'(z)=-2f(z)$
Find all analytic functions $f:\mathbb{C} \longrightarrow \mathbb{C}$ such that
$$f'(z)=-2f(z),~z \in \mathbb{C}$$ and
$$f(0)+f'(0)=1$$
the only thing that majorly concerns me is making sure my ...
- 1,013
2
votes
5
answers
821
views
Solutions in terms of the hypergeometric functions
Is it possible to somehow express the solutions to the differential equations:
$$\frac{d^2y}{dx^2} + \bigg(\frac{1}{x + 8} - \frac{1}{x} + \frac{1}{x - 1} + \frac{1}{x - 4}\bigg) \frac{dy}{dx} + \bigg(...
- 10.2k
4
votes
1
answer
363
views
How to relate the solutions to a Fuchsian type differential equation to the solutions to the hypergeometric differential equation?
Consider a Fuchsian type differential equation written as
$$\frac{d^2 y}{dz^2} + \frac{p(z)}{(z - z_1)(z - z_2) \cdots (z - z_m)} \frac{dy}{dz} + \frac{q(z)}{(z - z_1)^2 (z - z_2)^2 \cdots (z - z_m)^2}...
- 10.2k
0
votes
2
answers
132
views
Formal power series and complex ODE
I have this Ode $zS''+S'+zS=0$ $z\in\mathbb{C}$ and $S$ a 2 times differentiable function. I was looking into the formal power series $S(X)=\sum_{n\geq0}a_nX^n$ that verify this ODE and i'm supposed ...
- 33
2
votes
0
answers
861
views
Proving the Bessel function solves the Bessel equation
Using the notation for the Bessel function as $J_n(z)=\sum \limits_{k=0}^{\infty}\frac{(-1)^kz^{n+2k}}{k!(n+k)!2^{n+2k}}$, I want to show that $w=J_n(z)$ satisfies $w''+\frac{1}{z}w'+\left(1-\frac{n^2}...
- 1,397
1
vote
0
answers
78
views
A theoretical question regarding Frobenius method
The following is a theoretical question regarding Frobenius method. Let $b(x),c(x)$ be real functions analytic at 0. Let $b(x)=\sum_{i=0}^\infty b_ix^i, c(x)=\sum_{i=0}^\infty c_ix^i$ on $(-R,R)$. ...
- 14.3k
1
vote
1
answer
130
views
Prove $\sum_{m \geq 1} {\frac{(2m-2)!}{(1-\rho)\cdots(m-\rho)} \frac{t^m}{(1-x)^{2m-1}}} $is divergent
How do I show that the following power series is divergent?
$$ u(t,x) = \sum_{m \geq 1} {\dfrac{(2m-2)!}{(1-\rho)\cdots(m-\rho)} \dfrac{t^m}{(1-x)^{2m-1}}} $$
where $t$ is complex 1-dimensional, $x$ ...
- 942
0
votes
2
answers
57
views
a differential equation equation related to fourier series
I am really struggling with this one. Any help is welcome!
For equation $f''(z) + p(z) f'(z) + q(z) f(z) = 0$, where $p(z)$ and $q(z)$ are fixed polynomials.
Given $f(0)=f_0$, $f'(0)=f_1$, prove that ...
- 229
1
vote
1
answer
635
views
using power series expansion to find a holomorphic function which solves a differential equation
Using power series expansions,
find a function $f$ which is holomorphic on the unit disk $D:=$ {$z\in\mathbb C:|z|<1$}
and solves the differential equation
$(1-z^2)f''(z)-4zf'(z)-2f(z)=0$
for $...
- 311