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
0
votes
2
answers
30
views
Explain rewriting of expression
Can someone explain why $$\bigg(\frac{1}{5} - \frac{1}{10}\bigg)e^{(-1+2i)t}$$ is equal to writing $$\bigg(\frac{1}{5} - \frac{1}{10}\bigg)\bigg(e^{-t}\cos 2t+i e^{-t} \cdot \sin 2t \bigg)$$
- 605
0
votes
2
answers
115
views
multiplicative property of complex exponentials
How can I demonstrate the property $e^u\cdot e^v$ = $e^{u+v}$ for complex $u,v$ using the summation definition of $\exp(z)$. Specifically this is the definition saying that $\exp(z) = \sum_{k=0}^{\...
- 37
0
votes
1
answer
247
views
How to compute the exponential of this matrix?
I am trying to prove all the results regarding linear algebra in my ODE class. I have already convinced myself that if I have a matrix $T$ which has an eigenvalue $\lambda = a + ib$ and an associated ...
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 \...
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