公式

グルサの定理

$$f^{(n)}(a)=\frac{n!}{2\pi i}\int_C\frac{f(z)}{(z-a)^{n+1}}\,dz \qquad(n\geq 0)$$

留数定理

$$\begin{align*} \frac{1}{2\pi i}\int_C f(z)\,dz&=\sum_{a}\frac{1}{(n-1)!}\lim_{z\to a}\frac{d^{n-1}}{dz^{n-1}}(z-a)^nf(z)\\ &=\sum_{a}\text{Res}_{z=a}f(z) \end{align*}$$

フーリエ変換　(Fourier Transform)

$$\begin{align*} \mathcal{F}[f](\xi)&=\hat{f}(\xi)=\int_{-\infty}^{\infty}f(x)e^{-2\pi i x\xi}\,dx\\ f(x)&=\int_{-\infty}^{\infty}\hat{f}e^{2\pi i x \xi}\,dx \end{align*}$$

ラプラス変換　(Laplace Transform)

$$\begin{align*} \mathcal{L}(s)&=F(s)=\int_0^\infty f(t)e^{-st}\,dt\\ f(t)&=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}F(s)e^{st}\,ds \end{align*}$$

メリン変換　(Mellin Transform)

$$\begin{align*} \mathcal{M}[f](s)&=\varphi(s)=\int_{0}^{\infty}f(x)x^{s-1}\,dx\\ f(x)&=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\varphi(s)x^{-s}\,ds \end{align*}$$

ポアソンの和公式　(Poisson Sum)

$$\begin{align*} \sum_{n\in\mathbb{Z}}f(n)=\sum_{n\in\mathbb{Z}}\mathcal{F}[f](n) \end{align*}$$

分数階微分

$$\begin{align*} D^{-\alpha}f(x)=\frac{1}{\Gamma(\alpha)}\int_{0}^{x}(x-t)^{\alpha-1}f(t)\,dt \end{align*}$$

複素数階微分

$$\begin{align*} D^\nu F(x)=\int_{0}^{x}Y_{-\nu}(x-y)F(y)\,dy=\left\{\begin{array}{ll} \displaystyle\frac{1}{\Gamma(-\nu)}\int_0^x(x-y)^{-(\nu+1)}\theta(x-y)F(y)\,dy & (\nu\neq1,\ 2,\ 3,\ \cdots)\\ \displaystyle\int_0^x\delta^{(\nu)}(x-y)F(y)\,dy & (\nu=1,\ 2,\ 3,\ \cdots) \end{array}\right. \end{align*}$$

正規化積　(Regularized Product)

丼積