1
vote
0answers
24 views

Runge Kutta stability region for forward euler and explicit midpoint

The interval of absolute stability is the intersection of the region of absolute stability in the complex plane with the real axis.Show that Runge Kutta forward Euler and RK explicit midpoint have the ...
1
vote
0answers
52 views

$w''+w~ sin z=0$ solutions

I have $w''+w~\sin z=0$ and I want to sow that this equation has at least one solution of the form $e^{c z}v(z)$ where $v(z)$ is periodic My idea is to start by expressing $w_1(z+2\pi)$ and ...
2
votes
0answers
60 views

second order linear ode in the complex domain

Consider $w''(z)+p(z)w'(z)+q(z)=0$ where $p(z), q(z)$ are analytic for $R\le|z|<\infty$ for some fixed $R$. Now I want to prove using analytic continuation of the solutions that the ode has one ...
0
votes
0answers
20 views

Showing uniqueness of character identity

How would one show that any complex-valued C1 function satisfying the character identity must be of the form exp(cx) for c complex. Given a function f, it is said to satisfy the character identity if ...
2
votes
0answers
42 views

Imaginary part of Log laplacian

I'm confused about how to calculate $\nabla^2 \log z$, where $z=re^{i\theta}$ is a complex number. My calculations return $$ \nabla^2 \log z = 2\pi\frac{\delta(r)}{r} [\delta(\theta) + i ...
5
votes
1answer
114 views

Rank of a jet bundle of a vector bundle.

I am trying to understand the jet bundles but currently I am stuck on the following questions: Let $\pi: E\rightarrow X$ be a smooth (holomorphic) vector bundle of rank $k$ over a smooth (complex) ...
0
votes
1answer
43 views

Solution of this ODE

How to solve the following ODE? $$i\partial_tv=t^q|v|^pv$$ where $i$ is the imaginary unit and $v$ is complex valued? I think that separation of variables is to be used.
3
votes
1answer
70 views

Fourier series for e^x

I'm trying to teach myself partial differential equations from Strauss' book. I have run into a very bizarre problem - I cannot figure out what is the Fourier series of e^x! And not even Google has ...
3
votes
2answers
64 views

Justifying an ODE's solution

In an introductory lesson into ODEs, in order to "semi-rigorously" justify the solution for e.g. : $(a)\ \ y'+y=0$ we proceed without an ansatz or guess solution (hence the "semi-rigour"): Let: ...
1
vote
1answer
78 views

Recursive coefficients for an infinte series in complex analysis / differential equations

I have a question that feels rather simple, but I seem to be stumped! Given that $f$ is entire, use a power series representation of $f$ about $0$ to solve the differential equation ...
0
votes
1answer
25 views

Differentiable cauchy riemann equation

If f is differentiable and |f(z) = 7| in D(0,5) then f(z) is a constant function on this disk D(0,5). Is this true?
3
votes
4answers
274 views

If $f$ is holomorphic and $\,f'' = f$, then $f(z) = A \cosh z + B \sinh z$

Suppose $f$ is holomorphic in a disk centered at the origin and $f$ satisfies the differential equation $$f'' = f.$$ Show that $f$ is of the form $$f(z)=A \sinh z + B \cosh z,$$ for suitable constants ...
0
votes
2answers
71 views

A Complex Variable ODE

suppose $f$ is a holomorphic function on some domain $D$ satisfying $f'(z)=af(z) $ for some >constant a. show that $f(z)=Ce^{az}$, for some constant $C$
0
votes
1answer
53 views

Entire function and the Exponential function

Given an entire function $f : f(z)=f'(z) \forall z \in \Bbb C$, I need to prove that $\exists c \in \Bbb C : f(z)=ce^z$. Even though it seems very intuitive I could not prove it completely. My ...
0
votes
1answer
44 views

Show the differential solution

I'm a little rusty on Diff Eq. I remember the basic concepts but I'm confused where the log part comes from.
1
vote
0answers
24 views

How to linearize and solve ODE $\dot{z}_n = \sum_{m} i M_{nm} \frac{z_m - z_n}{|z_m - z_n|}$ for $z_n\approx 0$?

I came across a physical system which obeys the following ODE $$\frac{d z_n}{dt} = \sum_{m=1}^N i M_{nm} \frac{z_m - z_n}{|z_m - z_n|}, \qquad n\in\{1,2,\dots,N\}$$ where $z_n \equiv z_n(t)$ are ...
1
vote
0answers
37 views

transformation of an implicit function

If given an implicit function $$P(x) =\frac{Q(t)}{R(t)^3}+\frac{S(t)}{R(t)} $$ where $t=t(x)$ and has to satisfy this constraint $$\left(\frac{dt}{dx}\right)^2= \frac{1}{R(t)}$$ Is it possible to ...
1
vote
0answers
34 views

On a positive semidefinite infinite matrix coming from a ordinary differental equation

Let $F: B\rightarrow R$ be the real analytic real valued function defined on an open ball $B\subset C^2$ by: \begin{equation}%\label{diastcal} F(z)=\sum_{k=1}^\infty c_k\left((z_1+\bar ...
1
vote
1answer
95 views

Real and imaginary parts of function defined by DE

So I have this differential equation: $$ \frac{\text{d}}{\text{d} \theta}u(\theta, E) = P(E)(1 - u^2)\sqrt{E + \gamma^2 \ln(1-u^2)} \tag{1} $$ Where $E > 0$, $P(E)$ is a complicated function I ...
1
vote
1answer
96 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, $x$ ...
0
votes
2answers
46 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 ...
1
vote
0answers
298 views

Poisson Integral Formula

I'm looking at the following problem. Prove that if $h$ is harmonic on an open neighborhood of the disc $B(w,\rho)$, then for $0 \leq r < \rho, 0 \leq t < 2\pi$, $$h ...
1
vote
1answer
77 views

Bessel function with complex argument

So I understand that the bessel functions of the first kind are the ones that satisfy this equation: $$x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}+(x^2-\alpha^2)y = 0$$ and the result is a linear ...
0
votes
0answers
28 views

Finding 2D potential distribution given boundary conditions.

I came across a certain problem which I don't know how to solve. I'm looking for a few hints or suggested readings which would lead me to a solution. The problem is to solve for the potential ...
0
votes
3answers
125 views

top journals in analysis

as an undergraduate I find analysis as my favorite.I want to read journals regarding that. give me top 5 journals in analysis(real,complex)? top 5 journals in differential geometry? and generally some ...
1
vote
0answers
97 views

Monodromy Groups of Differential Equations

I have heard that monodromy groups and analytic continuation can be used to construct new solutions to a differential equation from a particular solution. What references (textbook, or papers) could I ...
1
vote
0answers
33 views

Particle in a Polya Vector field

For a given analytic function $H$ from $\mathbb{C}$ to $\mathbb{C}$, we define the Polya Vector Field to be $\bar{H}$. This then corresponds to a irrotational, conservative vector field on ...
1
vote
0answers
42 views

Existence and Uniqueness of complex ODE's

I'm wondering if there is a theorem for the existence and uniqueness of complex ODE's. If there is, would someone mind explaining the general breadth of the theorem and/or directions to an online ...
2
votes
1answer
85 views

I want to solve this differential equation in $ℂ$

I want to solve this differential equation in $ℂ$: $$(1-2^{1-s})f′(s)=2^{1-s}(ln2)f(s)$$ where $f$ is an analytic function for all $s=α+iβ∈ℂ$ with $0<α<1$ and has infinitely many zeros in the ...
0
votes
1answer
92 views

Harmonic Conjugate in Star Domain

I have been given that $u(x,y)$ is a harmonic function on a star shaped domain $D$. I have to show that it has harmonic conjugate $v(x,y)$ on same domain given up to additive constant by ...
2
votes
1answer
55 views

Complex Differential Equation: $f'(z)=bf(z) \iff f(z)=ae^{bz}$

Let $f\colon G\to\mathbb{C}$ be holomorphic on the domain $G\subseteq\mathbb{C}$ and $b\in\mathbb{C}$. Show that the two following statements are equivalent: 1) $f(z)=ae^{bz}$ on $G$ with ...
1
vote
2answers
71 views

Differential equation system with complex eigenvalues

I need to solve this equation $$ x'=x+y, y'=-2x+3y$$ I get the matrix rigor and the eigenvectors complex $2-i$ and $2+i$. When I try to apply the eigenvectors associated the solution for $x$ I ...
1
vote
0answers
102 views

Approximating the modified Bessel’s function with a sum of exponentials

I am looking for an approximation for modified Bessel’s function $I_\alpha(f(t))$ (specially $I_0(f(t))$ or at least $I_0(t)$) with a sum of exponential functions. I mean I want to approximate the ...
2
votes
3answers
47 views

Functions differentiable on {$z \in \mathbb{C}: 0 < |z| < 1$}

This is a past exam question from a Complex Analysis exam paper.. Prove or disprove that there exists a function $f$ differentiable on {$z \in \mathbb{C}: 0 <|z|<1$} such that $(i) ...
-3
votes
1answer
73 views

Are functions differentiable on $\mathbb{C}$

What is the method i need to use to find out if functions are differentiable on $\mathbb{C}$ Some examples of past exam questions on this are.. (i) $f(x+iy)=x^2 -y^2+x+1+i(2xy+y)$ ...
0
votes
1answer
36 views

we need to show $Re(p)=0$.

Could any one tell me how to solve this one? $p\in\mathbb{C}$,consider the diff equ $$u''-p^2u=0$$, if every solution of the diff equ satisfies $$\sup_{T>0}\frac{1}{2T}\int_{-T}^{T} |u(t)| ...
3
votes
2answers
80 views

Solution to functional equation

I have the following functional equation: $$f(a+b)=f^{n-1}(a)f(b)+f^{n-2}(a)f^{1}(b)+...+f(a)f^{n-1}(b)=\sum_{k=0}^{n-1}f^{n-1-k}(a)f^{k}(b)$$ where $a,b$ are complex and the function $f$ is an ...
1
vote
1answer
187 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 ...
0
votes
0answers
46 views

unique holomorphic solution

Show that for any locally defined holomorphic function $g$ at $0$ $\in \mathbb{C}$, there is a unique holomorphic solution $f$ of the equation $f'=g \circ f$ on $D(0,r)$ with $f(0)=0$ for any ...
1
vote
0answers
88 views

Whats the purpose : Hilbert's problems in measure space

This may sound a very newbie question, anyway I would like to ask here and to make it more clear for me. I've got an assignment to consider boundary problems in space of finite measures W, where the ...
0
votes
1answer
174 views

A change of variables in the euler equation

If someone could help me with the proposed change of variables, it would be greatly appreciated. Consider Euler's equation: $$z^2w'' + \alpha zw' + \beta w = 0$$ where $w$ is a function of $z$ and ...
1
vote
1answer
104 views

Why are solutions to the Loewner equation analytic?

Consider the Loewner differential equation $$\frac{\partial g_t(z)}{\partial t} = \frac{2}{g_t(z) - \sqrt{\kappa} B_t}$$ where $B_t$ is a 1-dimensional standard Brownian motion starting at the ...
0
votes
1answer
219 views

Harmonic Extension

Let be $u$ a harmonic function defined on an open set $\Omega \setminus \{p\} \subset \mathbb{C}$ of the complex plane. Show that if $u$ is bounded in a neighborhood of $p$ then $u$ admits a harmonic ...
2
votes
2answers
160 views

Laplace transform exercise

I found this on Priestley's Complex Analysis in the Laplace transforms bit. Suppose $f$ satisfies $f'(t)=f(kt)$ for $t>0$, where $0<k<1$ and $f(0)=1$. Prove that ...
4
votes
1answer
252 views

solution of Lagrange differential equation are square integrable

I was recently posing myself this question. Given the Lagrange DE $$[(1-x^2)u']'+\lambda u=0,$$ where $\lambda$ is a real parameter and $x\in[-1,1]$, it is well known that, if $\lambda=n(n+1)$ for ...
3
votes
0answers
61 views

Complex nonlinear differential equation

I have the following nonlinear differential equation: $$\ddot z(t)-\sin(z(t))=0$$ where $z(t)$ is a complex variable. The solution of the same equation with $z(t)$ real, is a function of Jacobi ...
2
votes
0answers
40 views

If $y(z) = C y_0(z) \int_w^z \frac{d\zeta}{y_0(\zeta)^2}$, what limit can we take in $C$ and $w$ to obtain $y(z) \to y_0(z)$?

This is Exercise 6.5 from Miller's Applied Asymptotic Analysis. The book shows that, given any solution $y_0(z)$ to the equation $$ y''(z)+f(z)y(z)=0, \tag{1} $$ a general solution is given by $$ ...
1
vote
0answers
51 views

Find K and $\rho$ such that $|f_s| \leq K \rho^{-s}$ , with $f(z) = f_0 + f_1 z + \dots + f_s z^{s} + \dots$

I've been given the following equation; $u''-\frac{f(z)}{z^2}u=0$ , with $f$ an analytic function:$\ \ $ $f(z) = f_0 + f_1 z + f_2 z^2 + \dots$ Using a substitution, of $u(z) = v(z)z^{\alpha}$ you ...
1
vote
0answers
76 views

How to compute the values of this function ? ( Fabius function )

How to compute the values of this function ? ( Fabius function ) It is said not to be analytic but $C^\infty$ everywhere. But I do not even know how to compute its values. Im confused. Here is the ...
4
votes
0answers
331 views

Confused by a proof in Rudin *Functional Analysis*

I am reading Rudin's Functional Analysis and got quite confused by his proof of Thm 8.5, that is, the existence of fundamental solutions for differential operator $P(D)$, where $P$ is a polynomial. ...