Wikidot 當中的 LaTex 數學式 (範例集)

# 基本範例

[$] f(n) = \sum^{N-1}_{k=0} F(k) e^{i 2 \pi k} \frac{n}{N} [$]

(1)
\begin{align} f(n) = \sum^{N-1}_{k=0} F(k) e^{i 2 \pi k} \frac{n}{N} \end{align}

# 基本範例

[$] e = lim_{n \rightarrow \infty}\; 1+\frac{1}{1!}+\frac{1}{2!}+...+\frac{1}{n!} [$]

(2)
\begin{align} e = lim_{n \rightarrow \infty}\; 1+\frac{1}{1!}+\frac{1}{2!}+...+\frac{1}{n!} \end{align}

# 基本範例

[$] e = lim_{n \rightarrow \infty} (1+\frac{1}{x})^x [$]

(3)
\begin{align} e = lim_{n \rightarrow \infty} (1+\frac{1}{x})^x \end{align}

# 複雜範例

[$] \begin{split} \mathbf{T n} &= \left[T_{ij} \mathbf{e}_i \otimes \mathbf{e}_j \right] n_k \mathbf{e}_k \\ & = T_{ij} n_k \left(\mathbf{e}_i \otimes \mathbf{e}_j\right) \mathbf{e}_k \\ & = T_{ij} n_j \mathbf{e}_i \end{split} [$]

(4)
\begin{split} \mathbf{T n} &= \left[T_{ij} \mathbf{e}_i \otimes \mathbf{e}_j \right] n_k \mathbf{e}_k \\ & = T_{ij} n_k \left(\mathbf{e}_i \otimes \mathbf{e}_j\right) \mathbf{e}_k \\ & = T_{ij} n_j \mathbf{e}_i \end{split}

# 多行範例

[[math type="eqnarray"]]
f'(x) & = & \frac{d f(x)}{dx} = c_1+c_2*2*x+c_3*3*x^2+c_4*4*x^3+... \\
f''(x) & = & \frac{d f'(x)}{dx}  = c_2*2*1+c_3*3*2*x+c_4*4*3*x^2+... \\
f'''(x) & = & \frac{d f''(x)}{dx} = c_3*3*2*1+c_4*4*3*2*x+... \\
... \\
f^k(x) & = & \frac{d f^{k-1}(x)}{dx} = c_k k!+c_{k+1} (k+1)! x+...
[[/math]]

(5)
\begin{eqnarray} f'(x) & = & \frac{d f(x)}{dx} = c_1+c_2*2*x+c_3*3*x^2+c_4*4*x^3+... \\ f''(x) & = & \frac{d f'(x)}{dx} = c_2*2*1+c_3*3*2*x+c_4*4*3*x^2+... \\ f'''(x) & = & \frac{d f''(x)}{dx} = c_3*3*2*1+c_4*4*3*2*x+... \\ ... \\ f^k(x) & = & \frac{d f^{k-1}(x)}{dx} = c_k k!+c_{k+1} (k+1)! x+... \end{eqnarray}

# 矩陣範例

[$] \left[ \begin{array}{ccc} T_{11} & T_{12} & T_{13} \\ T_{21} & T_{22} & T_{23} \\ T_{31} & T_{32} & T_{33} \end{array} \right] [$]

(6)
\begin{align} \left[ \begin{array}{ccc} T_{11} & T_{12} & T_{13} \\ T_{21} & T_{22} & T_{23} \\ T_{31} & T_{32} & T_{33} \end{array} \right] \end{align}

# 矩陣範例

[$] \left\{ \begin{array}{c} t_1 \\ t_2 \\ t_3 \end{array} \right\} = \left[ \begin{array}{ccc} T_{11} & T_{12} & T_{13} \\ T_{21} & T_{22} & T_{23} \\ T_{31} & T_{32} & T_{33} \end{array} \right] \left\{ \begin{array}{c} n_1 \\ n_2 \\ n_3 \end{array} \right\} [$]

(7)
\begin{align} \left\{ \begin{array}{c} t_1 \\ t_2 \\ t_3 \end{array} \right\} = \left[ \begin{array}{ccc} T_{11} & T_{12} & T_{13} \\ T_{21} & T_{22} & T_{23} \\ T_{31} & T_{32} & T_{33} \end{array} \right] \left\{ \begin{array}{c} n_1 \\ n_2 \\ n_3 \end{array} \right\} \end{align}

# 範例

(8)
\begin{align} \int_1^x \frac{1}{x} dx = 1 \end{align}

# 範例

(9)
\begin{align} \frac{d}{dx} e^x = e^x \end{align}

# 範例

(10)
\begin{align} e^x = 1+\frac{1}{1!} x + \frac{2}{2!} x^2 + ... \frac{n}{n!} x^n+ ... \end{align}

# 範例

(11)
$$e^{i x} = cos(x) + i*sin(x)$$

# 範例

(12)
\begin{align} f(x) = c_0 + c_1 x + c_2 x^2 + ...+ c_k x^k+...=\sum_{k=0}^\infty c_k x^k \end{align}

# 範例

(13)
\begin{align} c_k = \frac{f^k(0)}{k!} \end{align}

# 範例

(14)
\begin{align} f(x) = f(0) + \frac{f'(0)}{1!} x +...+ \frac{f^k (0)}{k!} x^k+...=\sum^{\infty}_{k=0} \frac{f^k(0)}{k!} x^k \end{align}

# 範例

(15)
\begin{align} f(x) = f(a) + \frac{f'(a)}{1!} x +...+ \frac{f^{k(a)}}{k!} x^k+...= \sum^\infty_{k=0} \frac{f^k(a)}{k!} x^k \end{align}

# 範例

(16)
\begin{align} e^{i x} = 1 + i \frac{x}{1!} - \frac{x^2}{2!} - i \frac{x^3}{3!} + ... \end{align}

# 範例

(17)
\begin{align} cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} + ... \end{align}

# 範例

(18)
\begin{align} sin(x) = \frac{x}{1!} - \frac{x^3}{3!} + \frac{x^5}{5!} + ... \end{align}

# 範例

(19)
$$e^{i x} = cos(x) + i * sin(x)$$

# 範例

(20)
\begin{align} f(x) = \frac{a_0}{2} + \sum^{\infty}_{n=- \infty} a_n cos(n x)+ b_n sin(n x) \end{align}

# 範例

(21)
$$cos(n x) + i * sin(n x) = e^{i n x}$$

# 範例

(22)
\begin{align} f(x) = \sum^{\infty}_{n=-\infty} F_n e^{i n x} \end{align}

# 範例

(23)
\begin{align} F_t = \frac{1}{2\pi} \int^{\pi}_{-\pi} f(x) e^{i t x} dx \end{align}

# 範例

 $F_0 = \frac{1}{2}$ $F_n= \frac{1}{2} (a_n- i b_n)$ $F_{- n}= \frac{1}{2} (a_n+i b_n)$ $a_0 = 2 c_0$ $a_n=F_n+F_{- n}$ $b_n=i (F_n-F_{-n})$

# 範例

(24)
\begin{align} f(t) = \int^\infty_{- \infty} F(x) e^{i 2 \pi x t} dt \end{align}

# 範例

\begin{eqnarray}
f(x) = \left\{
\begin{array}{c}
1 \qquad x \in Z \\
\end{eqnarray}