# Questions tagged [laplace-transform]

The Laplace transform is a widely used integral transform (transformation of functions by integrals), similar to the Fourier transform.

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### Complex Fraction magnitude

I'm having a problem with finding the magnitude of the transfer function. $$G(s) = \frac{s+20}{(s+1)(s+100)}$$ I know that if we have a simple complex in the rectangular form $$Z = a + jb$$ The ...
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### simple pendulum equation, why it cannot be solved with laplace transform (the general solution)

Usually to solve the simple pendulum equation: $\qquad \ell {\ddot \theta }+g\sin \theta =0\,$ Using the first term of Taylor series is used as approximation, but although $\sin \theta$ can be ...
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### Find a sequence $(b_n)_n$ such that $\sum_n b_n = 0$ and $g(t) \equiv \sum_n e^{-2\lambda_n t}a_n^2 + \sum_n e^{-\lambda_n t} a_n b_n + \sum_n b_n^2$

Fix a sequence of real numbers $(a_n)_n \in \ell^2$ and a sequence of positive numbers $(\lambda_n)_n \in \ell^2$. Question. What is a crisp characterization of functions $g:[0,\infty) \to [0,\infty)$,...
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### Simplification Error Inverse Laplace Transform

I am trying to build a model of a electronic circuit and to solve the differential equation using the laplace transformation. With the help of numeric simulation I know what one factor has to be and ...
Let $W_0=p\delta_0 + (1-p)\text{exponential}(1-p)$, where $0<p<1$, $\delta_0$ is the point mass at zero and exponential($\cdot$) is the exponential distribution. How do I calculate the Laplace ...