Tagged Questions

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 ...
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 ...
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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): $$F_{2l}(x)=x\, \tanh(\pi x/2)\left|\int_0^1 u^{i x-1}P_{2l}(u)\,du\right|^2$$ ...
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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 ...
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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 ...
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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) ...
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.
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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 ...
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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: ...
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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 ...
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?
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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 ...
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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 $C$
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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 ...
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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.
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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 ...
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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: %\label{diastcal} F(z)=\sum_{k=1}^\infty c_k\left((z_1+\bar ...
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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 ...
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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, $x$ ...
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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 ...
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 $0 \leq r < \rho, 0 \leq t < 2\pi$, $$h ... 0answers 62 views Inverse Laplace Transform using Jordan's Lemma? Following is the question that i am trying to solve: "Consider a second order linear ODE x\dfrac{d^2y}{dx^2}+x\dfrac{dy}{dx}+(3-2x)y=0 A) Find the solution employing Laplace integrals by ... 1answer 103 views Bessel function with complex argument So I understand that the bessel functions of the first kind are the ones that satisfy this equation:$$x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}+(x^2-\alpha^2)y = 0$$and the result is a linear ... 0answers 33 views Finding 2D potential distribution given boundary conditions. I came across a certain problem which I don't know how to solve. I'm looking for a few hints or suggested readings which would lead me to a solution. The problem is to solve for the potential ... 3answers 140 views top journals in analysis as an undergraduate I find analysis as my favorite.I want to read journals regarding that. give me top 5 journals in analysis(real,complex)? top 5 journals in differential geometry? and generally some ... 0answers 111 views Monodromy Groups of Differential Equations I have heard that monodromy groups and analytic continuation can be used to construct new solutions to a differential equation from a particular solution. What references (textbook, or papers) could I ... 0answers 44 views Particle in a Polya Vector field For a given analytic function H from \mathbb{C} to \mathbb{C}, we define the Polya Vector Field to be \bar{H}. This then corresponds to a irrotational, conservative vector field on ... 0answers 53 views Existence and Uniqueness of complex ODE's I'm wondering if there is a theorem for the existence and uniqueness of complex ODE's. If there is, would someone mind explaining the general breadth of the theorem and/or directions to an online ... 1answer 87 views I want to solve this differential equation in ℂ I want to solve this differential equation in ℂ:$$(1-2^{1-s})f′(s)=2^{1-s}(ln2)f(s)$$where f is an analytic function for all s=α+iβ∈ℂ with 0<α<1 and has infinitely many zeros in the ... 1answer 99 views Harmonic Conjugate in Star Domain I have been given that u(x,y) is a harmonic function on a star shaped domain D. I have to show that it has harmonic conjugate v(x,y) on same domain given up to additive constant by ... 1answer 64 views Complex Differential Equation: f'(z)=bf(z) \iff f(z)=ae^{bz} Let f\colon G\to\mathbb{C} be holomorphic on the domain G\subseteq\mathbb{C} and b\in\mathbb{C}. Show that the two following statements are equivalent: 1) f(z)=ae^{bz} on G with ... 2answers 74 views Differential equation system with complex eigenvalues I need to solve this equation$$ x'=x+y, y'=-2x+3y$$I get the matrix rigor and the eigenvectors complex 2-i and 2+i. When I try to apply the eigenvectors associated the solution for x I ... 0answers 106 views Approximating the modified Bessel’s function with a sum of exponentials I am looking for an approximation for modified Bessel’s function I_\alpha(f(t)) (specially I_0(f(t)) or at least I_0(t)) with a sum of exponential functions. I mean I want to approximate the ... 3answers 50 views Functions differentiable on {z \in \mathbb{C}: 0 < |z| < 1} This is a past exam question from a Complex Analysis exam paper.. Prove or disprove that there exists a function f differentiable on {z \in \mathbb{C}: 0 <|z|<1} such that (i) ... 1answer 73 views Are functions differentiable on \mathbb{C} What is the method i need to use to find out if functions are differentiable on \mathbb{C} Some examples of past exam questions on this are.. (i) f(x+iy)=x^2 -y^2+x+1+i(2xy+y) ... 1answer 36 views we need to show Re(p)=0. Could any one tell me how to solve this one? p\in\mathbb{C},consider the diff equ$$u''-p^2u=0$$, if every solution of the diff equ satisfies$$\sup_{T>0}\frac{1}{2T}\int_{-T}^{T} |u(t)| ...
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 ...