2
votes
2answers
115 views

1st order separable ODE involving the complex conjugate of the dependent variable

Is there a closed form (complex) solution $z(t)$ to the equation \begin{align} \frac{dz}{dt}=f(t)\bar{z}, \end{align} (the bar means complex conjugate) for any given complex valued function $f$ of a ...
1
vote
1answer
43 views

Solution to second order differential equation

I'm reading a paper in which the authors solve the following equation: $\frac{d^{2}}{dz^{2}}\hat{p}$($\bf{q}$$,z)$-$q^{2}\hat{p}$($\bf{q}$$,z)$-$\frac{iq_{y}}{(2\pi)^{2}}\delta(z-z_{2})$=0 here ...
6
votes
0answers
65 views

Proving that two functions involving integrals with Legendre polynomials are equal

I have two functions that I expect to be equal (where $P_{2l}$ are the even Legendre Polynomials): $$F_{2l}(x)=x\, \tanh(\pi x/2)\left|\int_0^1 u^{i x-1}P_{2l}(u)\,du\right|^2$$ ...
0
votes
2answers
24 views

What are the complex solutions of a linear homogenous ODE of order $n$ with constant coefficients?

What are the complex solutions of a linear homogenous ODE of order $n$ with constant coefficients? Where can I read a proof? p.s. I don't even see the answer to the first question with a google ...
2
votes
0answers
37 views

Functional Equation involving derivatives and time-steps [duplicate]

I am attempting to solve the equation $$f(x + 1) = f'(x)$$ for distributions $C \rightarrow C: f(x)$ My first guess to exploit the fact that this seems similar to identity $$\sin\left( ...
3
votes
1answer
136 views

Solving the ODE $[(1-x^2)\frac{\partial}{\partial x} - \lambda]f = [\frac{\partial}{\partial x} - \frac{\lambda}{a}]g$

I want to solve $f(x)$ in terms of $g(x)$ in the following ODE $$\left[(1-x^2)\frac{\partial}{\partial x} - \lambda\right]f(x) = \left[\frac{\partial}{\partial x} - \frac{\lambda}{a}\right]g(x),$$ ...
0
votes
1answer
51 views

How to solve these two differential equation?

I try to solve these two difference equation ; $$ \frac{dq}{dz} = -j\left(b_1q - kp\right),\\ \frac{dp}{dz} = -j\left(b_2p - kq\right) $$ where $j$ stands for $\sqrt{-1}$, and $b_1$ ,$b_2$ and k are ...
1
vote
0answers
27 views

Uniqueness of holomorphic solutions of a differential equation

Given two polynomials $p,q\in\mathbb C[z]$ consider the initial value problem \begin{align*} f(z)-p(z)f'(z)&=f(z^2)-q(z)f'(z^2),\qquad z\in\mathbb D,\\ f(0)&=0, \\ f'(0)&=1. \end{align*} ...
1
vote
0answers
25 views

Linear homogeneous ODE system of first order

Good afternoon. I recently encountered the following problem to which I couldn't find a solution anywhere so far: Given $A:D\to\mathbb C^{2\times 2}$, $D\subset\mathbb C$ open, with holomorphic ...
1
vote
1answer
77 views

Does $\int_{-\infty}^{\infty}{\frac{\mathrm{exp}(-t^2)}{t-iz} dt}=i \sqrt{\pi} e^{z^2} \mathrm{erfc}(z)$ hold for all $z$?

I have been working on a calculation that involves the following type of integral: $$ f(z)={\frac{1}{i\sqrt{\pi}}}\int_{-\infty}^{\infty}{\frac{e^{-t^2}}{t-iz} dt} \hspace{1.5cm} z \in \Bbb{C} ...
3
votes
1answer
83 views

Monodromy representation of Airy equation

Let $K=\Bbb{C}(z)$ with the usual derivation and consider the Airy dierential equation $y^{(2)}-zy$=0. How to determine the monodromy representration? Airy equation is not Fuchsian diferential ...
7
votes
3answers
113 views

Integral formulation for the solution of $xy'' + y' = y$

Let's say that $y$ satisfies the following ODE: $$xy'' + y' = y$$ I want to formulate $y$ as a contour integral. I know that the final result I should get is: $$y(x)=\frac{1}{2i\pi} ...
2
votes
2answers
76 views

How to solve this system of equations that appears in a ODE exercise?

I am trying to solve this equation, we know $A, B, Q,\phi\in\mathbb{R}$. \begin{eqnarray} T''(x) &=& \phi (T(x)-Q) \\ T(0)&=& A\\ T(b)&=&B \end{eqnarray} So the ...
1
vote
1answer
36 views

An exercise in Treves related to Cauchy-Riemann operator

This is part of the exercise 5.10 from the book "basic linear partial differential equations" by Treves: " Let $P(z)$ be a polynomial in one variable, with complex coefficients. Describe all ...
4
votes
4answers
139 views

A question regarding Frobenious method in ODE

Suppose $b(x),c(x)$ are 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 $(-R,R)$. Suppose $r$ is a double root of $r(r-1)+b_0r+c_0=0$. It is well ...
1
vote
0answers
45 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 $(-R,R)$. ...
1
vote
0answers
47 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 ...
2
votes
0answers
44 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 ...
6
votes
1answer
128 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
50 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
101 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
72 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
87 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
26 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
376 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
75 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
61 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
45 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
27 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

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
96 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
98 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
47 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
379 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
0answers
62 views

Inverse Laplace Transform using Jordan's Lemma?

Following is the question that i am trying to solve: "Consider a second order linear ODE $x\dfrac{d^2y}{dx^2}+x\dfrac{dy}{dx}+(3-2x)y=0$ A) Find the solution employing Laplace integrals by ...
1
vote
1answer
103 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
33 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
140 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
111 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
44 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
53 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
87 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
99 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
64 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
74 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
106 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
50 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 ...