All Questions
6
questions
0
votes
1
answer
358
views
Second order differential equation with complex coefficient
I have some doubts about this kind of second - order differential equation, which is used a lot in physics and for which there are many topics (but in this case the situation is a bit different ...
- 239
1
vote
2
answers
111
views
Solving $y'' + 2y' + 2y = 0$: How to eliminate imaginary unit from solution?
$$y'' + 2y' + 2y = 0$$
$\downarrow$ (write characteristic equation)
$\lambda^2 +2\lambda + 2 = 0$
$\downarrow$ (solve characteristic equation)
$\lambda = -1 \pm i$
$\downarrow$ (write general ...
- 1,922
1
vote
1
answer
2k
views
Proof of existence and uniqueness of the exponential function using ODEs
In our lecture notes for our complex analysis class, we were given the following theorem:
Theorem: There exists a unique complex function $f$ such that
$f(z)$ is a single valued function $f(z) \in \...
user100463
1
vote
4
answers
2k
views
Prove $f'(z)=f(z)$ implies $f(z)=ce^z$ for some $c \in \mathbb{C}$ [duplicate]
Suppose that: $$f'(z)=f(z) \text{ for all }z\in\mathbb C.$$ In other words, the complex function $f$ is equal to its own derivative. Prove that there is a constant $c\in\mathbb C$ such that $f(z)=c e^...
- 3,430
2
votes
0
answers
61
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( \frac{\pi}{...
- 17.5k
0
votes
2
answers
197
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 $C$
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