Fourier and Laplace Transform, and Differential Equations
1. Fourier series
\[f(x) = a_0 + \sum_{n=1}^{\infty} \left( a_n\cos \frac{n\pi}{L} + b_n\sin \frac{n\pi}{L}x\right).\] where
- a0 : \[a_0 = \frac{1}{2L}\int_{-L}^Lf(t)\mathrm{d}t.\]
- an : \[\frac{1}{L}\int_{-L}^L f(t)\cos \frac{n\pi}{L}t\mathrm{d}t\]
- bn : \[\frac{1}{L}\int_{-L}^L f(t)\sin \frac{n\pi}{L}t\mathrm{d}t\]
1.1. Half Fourier series
\[\phi(x)\sum_{n=1}^{\infty} C_n\sin \frac{n\pi x}{L}, \;C_n = \frac{2}{L}\int_0^L\phi(x)\sin \frac{n\pi x}{L}\mathrm{d}x.\]
\[\phi(x) = D_0 + \sum_{n=1}^{\infty}D_n \cos \frac{n\pi x}{L}, \;D_0 = \frac{1}{L}\int_0^L\phi(x)\mathrm{d}x, \;D_n = \frac{2}{L}\int_0^L\phi(x)\cos \frac{n\pi x}{L}\mathrm{d}x.\]
1.2. Fourier Integration
\[f(x) = \int_0^{\infty} \left[ A(\omega)\cos\omega x + B(\omega)\sin\omega x \right]\mathrm{d}\omega. \] Where \[A(\omega) = \frac{1}{\pi}\int_{-\infty}^{\infty}f(t)\cos\omega t \mathrm{d}t, \;B(\omega) = \frac{1}{\pi}\int_{-\infty}^{\infty}f(t)\sin\omega t \mathrm{d}t.\]
1.3. Fourier transform
- \( -\infty < \omega < \infty \) \[F(\omega) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} f(x)\exp(-\mathrm{i}\omega x)\mathrm{d}x. \]
- \(-\infty < x \infty\) \[f(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} F(\omega)\exp(\mathrm{i}\omega x)\mathrm{d}\omega, \]
\(F(\omega) = \mathcal{F}[f(x)]\), \(f(x) = \mathcal{F}^{-1}[F(\omega)]\). \(\Rightarrow F(\omega)\leftrightarrow f(x)\). \[\Rightarrow F(0) = \int_{-\infty}^{\infty}f(x)\mathrm{d}x. \; f(0) = \frac{1}{2\pi}\int_{-\infty}^{\infty}F(\omega)\mathrm{d}\omega.\] Also \(\mathcal{F}(C_1f_1 + C_2f_2) = C_1\mathcal{F}(f_1) + C_2\mathcal{F}(f_2)\).
\[\frac{\mathrm{d}f(x)}{\mathrm{d}x} \leftrightarrow \mathrm{i}\omega F(\omega) \Rightarrow f^{(n)}(x) \leftrightarrow (\mathrm{i}\omega)^n F(\omega).\]
\[xf(x)\leftrightarrow \mathrm{i}\frac{\mathrm{d}F(\omega)}{\mathrm{d}\omega} \Rightarrow x^n f(x)\leftrightarrow \mathrm{i}^n \frac{\mathrm{d}F(\omega)}{\mathrm{d}\omega}. \]
\[\int_{x_0}^xf(x)\mathrm{d}x\leftrightarrow\frac{F(\omega)}{\mathrm{i}\omega}.\]
\[f(x + \xi) = \exp(\mathrm{i}\omega\xi)F(\omega)\]
\[f_1(x)\ast f_2(x) = \int_{\infty}^{\infty}f_1(\xi)f_2(x-\xi)\mathrm{d}\xi, \;f_1(x)\ast f_2(x) = F_1(\omega)F_2(\omega). \]
Generic function \(\delta(x - x_0) = \)
- 0 when x ≠ x0,
- ∞ when x = x0.
Also \[\int_{-\infty}^{\infty}\delta(x - x_0)\mathrm{d}x = 1.\] \[\Rightarrow \int_{-\infty}^{\infty}f(x)\delta(x - x_0)\mathrm{d}x = f(x_0). \;\delta(x - a)\ast f(x) = f(x - a). \] Hence \( \delta(x -a)\ast \delta(x - b) = \delta[x - (a + b)]. \) \[\int_{-\infty}^{\infty} g(t)u(t)\exp(-\beta t) \exp(-\mathrm{i}\omega t)\mathrm{d}t = \int_0^{\infty}f(t)\exp(-pt)\mathrm{d}t.\] Where \(p = \beta + \mathrm{i}\omega, f(t) = g(t)u(t)\).
2. Laplace transform
\(\Re p = \beta > 0\):
- \[F(p) = \int_0^{\infty} f(t)\exp(-pt) \mathrm{d}t \]
- \[f(t) = \frac{1}{2\pi}\lim_{T\rightarrow\infty} \int_{\beta-\mathrm{i}T}^{\beta+\mathrm{i}T} F(p)\exp(pt)\mathrm{d}p.\]
\[\Rightarrow F(p) = \mathrm{L}[f(t)], F(p)\leftrightarrow f(t).\]
\[\mathcal{L}(C_1f_1 + C_2f_2) = C_1\mathcal{L}(f_1) + C_2\mathcal{L}(f_2).\]
\[\frac{\mathrm{d}f(t)}{\mathrm{d}t}\leftrightarrow pF(p) - f(0).\]
\[tf(t) \leftrightarrow -\frac{\mathrm{d}F(p)}{\mathrm{d} p}.\]
\[\int_0^t f(t)\leftrightarrow \frac{F(p)}{p}.\]
\[\exp a t f(t) \leftrightarrow F(p - a).\]
\[u(t-a)f(t-a)\leftrightarrow\exp(-a p) F(p).\] where \(u(t -a) =\)
- 0 when \(t < a\)
- \(f(t - a)\) when \(t \le a\).
\[f_1(t)\ast f_2(t)\leftrightarrow F_1(p)F_2(p).\]