ece205: add periodicity

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eggy 2023-11-07 13:06:00 -05:00
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@ -214,6 +214,38 @@ Thus we also have:
$$\mathcal L\{\delta (t-a)\}=e^{-as}\implies\mathcal L^{-1}\{1\}=\delta(t)$$ $$\mathcal L\{\delta (t-a)\}=e^{-as}\implies\mathcal L^{-1}\{1\}=\delta(t)$$
## Heat flow
The temperature of a tube from $x=0$ to $x=L$ can be represented by the following DE:
$$\text{temp}=u(x,t)=\boxed{u_t=a^2u_{xx}},0<x<L,y>0$$
Two boundary conditions are requred to solve the problem for all $t>0$ — that at $t=0$ and at $x=0,x=L$.
- $u(x,0)=f(x),0\leq x\leq L$
- e.g., $u(0,t)=u(L,t)=0,t>0$
### Periodicity
The **period** of a function is an increment that always returns the same value: $f(x+T)=f(x)$, and its **fundamental period** of a function is the smallest possible period.
!!! example
The fundamental period of $\sin x$ is $2\pi$, but any $2\pi K,k\in\mathbb N$ is a period.
The fundamental periods of $\sin \omega x$ and $\cos\omega x$ are both $\frac{2\pi}{\omega}$.
If functions $f$ and $g$ have a period $T$, then both $af+bg$ and $fg$ also must have period $T$.
#### Manipulating polarity
- even: $\int^L_{-L}f(x)dx=2\int^L_0f(x)dx$
- odd: $\int^L_{-L}f(x)dx=0$
- even × even = even
- odd × odd = even
- even × odd = odd
## Resources ## Resources
- [Laplace Table](/resources/ece/laplace.pdf) - [Laplace Table](/resources/ece/laplace.pdf)