\[ \sum_{\forall n \in \mathbb{N}_+}\frac1n = \prod_{\forall p\text{: 素数}}\frac1{1-\frac1p} \]
\[ \sum_{\forall m\text{: 奇数}}\frac{(-1)^\frac{m-1}2}m = \prod_{\forall p\text{: 奇数の素数}}\frac1{1 - \frac{(-1)^\frac{p-1}2}p} = \sum_{\forall p\text{: 奇数の素数}}\frac{(-1)^\frac{p-1}2}p = \frac\pi4 \]
\[ \sum_{\forall n \in \mathbb{N}_+}\frac1{n^2} = \frac{\pi^2}6 \]
\[ \sum_{\forall n \in \mathbb{N}_+}n = -\frac1{12} \]
\[ \sum_{\forall n \in \mathbb{N}_+}\frac1{n^s} = \prod_{\forall p\text{: 素数}}\frac1{1-\frac1{p^s}} \]
\[ \sum_{\forall n \in \mathbb{N}_0}\frac1{n!} = e \]
\[ e = 2 + \frac1{1 + \frac1{2 + \frac1{1 + \frac1{1 + \frac1{4 + \frac1{1 + \frac1{1 + \frac1{6 + \frac1{1 + \frac1{1 + \frac1{8 + \frac1{1 + \frac1{1 + \frac1{10 + \ddots}}}}}}}}}}}}}} \]
\[ e = 2 + \frac{2}{2 + \frac{3}{3 + \frac{4}{4 + \frac{5}{5 + \frac{6}{6 + \frac{7}{7 + \frac{8}{8 + \frac{9}{9 + \frac{10}{10 + \ddots}}}}}}}}} \]
\[ \pi = \frac{4}{1 + \frac{1^2}{3 + \frac{2^2}{5 + \frac{3^2}{7 + \frac{4^2}{9 + \frac{5^2}{11 + \ddots}}}}}} \]
\[ e^{i\pi} + 1 = 0 \]
\[ e^{i\theta} = \cos\theta + i\sin\theta \]
\[ i^2 = -1, \] \[ A = a + ib = \begin{pmatrix} a & -b\\ b & a \end{pmatrix} ,\quad C = c + id = \begin{pmatrix} c & -d\\ d & c \end{pmatrix} ,\quad \bar{A} = a - ib = \begin{pmatrix} a & b\\ -b & a \end{pmatrix} ,\quad \left|A\right|^2 = A\bar{A}, a, b, c, d\in\mathbb{R} \]
\[ A + C = a + ib + (c + id) = \begin{pmatrix} a & -b\\ b & a \end{pmatrix} + \begin{pmatrix} c & -d\\ d & c \end{pmatrix} \]\[ = a + c + i(b + d) = \begin{pmatrix} a + c & -(b + d)\\ b + d & a + c \end{pmatrix} \]
\[ A - C = a + ib - (c + id) = \begin{pmatrix} a & -b\\ b & a \end{pmatrix} - \begin{pmatrix} c & -d\\ d & c \end{pmatrix} \]\[ = a - c + i(b - d) = \begin{pmatrix} a - c & -(b - d)\\ b - d & a - c \end{pmatrix} \]
\[ AC = (a + ib)(c + id) = \begin{pmatrix} a & -b\\ b & a \end{pmatrix} \begin{pmatrix} c & -d\\ d & c \end{pmatrix} \]\[ = ac - bd + i(ad + bc) = \begin{pmatrix} ac - bd & -(ad + bc)\\ ad + bc & ac - bd \end{pmatrix} \]
\[ A\bar{A} = (a + ib)(a - ib) = \begin{pmatrix} a & -b\\ b & a \end{pmatrix} \begin{pmatrix} a & b\\ -b & a \end{pmatrix} = \begin{pmatrix} a^2 + b^2 & 0\\ 0 & a^2 + b^2 \end{pmatrix} \]\[ = a^2 + b^2 = (a^2 + b^2) \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \]
where \(A\neq 0\), \(\left|A\right|^2\neq 0\)\[ A^{-1} = \frac{\bar{A}}{\left|A\right|^2} = \frac{a - ib}{\left|A\right|^2} = \frac1{\left|A\right|^2} \begin{pmatrix} a & b\\ -b & a \end{pmatrix} \]
\[ i^2 = j^2 = k^2 = -1,\quad ij = -ji = k,\quad jk = -kj = i,\quad ki = -ik = j, \] \[ \iota^2 = -1, \] \[ A = a+bi+cj+dk = a \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} + b \begin{pmatrix} \iota & 0\\ 0 & -\iota \end{pmatrix} + c \begin{pmatrix} 0 & -1\\ 1 & 0 \end{pmatrix} + d \begin{pmatrix} 0 & -\iota\\ -\iota & 0 \end{pmatrix} = \begin{pmatrix} a + \iota b & -(c + \iota d)\\ c - \iota d & a - \iota b \end{pmatrix} = \begin{pmatrix} \alpha & -\beta\\ \bar\beta & \bar\alpha \end{pmatrix} , \] \[ A_2 = a_2+b_2i+c_2j+d_2k = a_2 \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} + b_2 \begin{pmatrix} \iota & 0\\ 0 & -\iota \end{pmatrix} + c_2 \begin{pmatrix} 0 & -1\\ 1 & 0 \end{pmatrix} + d_2 \begin{pmatrix} 0 & -\iota\\ -\iota & 0 \end{pmatrix} = \begin{pmatrix} a_2 + \iota b_2 & -(c_2 + \iota d_2)\\ c_2 - \iota d_2 & a_2 - \iota b_2 \end{pmatrix} = \begin{pmatrix} \gamma & -\delta\\ \bar\delta & \bar\gamma \end{pmatrix} , \] \[ \bar{A} = a-bi-cj-dk = a \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} - b \begin{pmatrix} \iota & 0\\ 0 & -\iota \end{pmatrix} - c \begin{pmatrix} 0 & -1\\ 1 & 0 \end{pmatrix} - d \begin{pmatrix} 0 & -\iota\\ -\iota & 0 \end{pmatrix} = \begin{pmatrix} a - \iota b & c + \iota d\\ -(c - \iota d) & a + \iota b \end{pmatrix} = \begin{pmatrix} \bar\alpha & \beta\\ -\bar\beta & \alpha \end{pmatrix} ,\quad \left|A\right|^2 = A\bar{A}, \alpha, \beta, \gamma, \delta\in\mathbb{C} \]
\[ A + A_2 = a+bi+cj+dk + (a_2+b_2i+c_2j+d_2k) = \begin{pmatrix} a + \iota b & -(c + \iota d)\\ c - \iota d & a - \iota b \end{pmatrix} + \begin{pmatrix} a_2 + \iota b_2 & -(c_2 + \iota d_2)\\ c_2 - \iota d_2 & a_2 - \iota b_2 \end{pmatrix} = \begin{pmatrix} \alpha & -\beta\\ \bar\beta & \bar\alpha \end{pmatrix} + \begin{pmatrix} \gamma & -\delta\\ \bar\delta & \bar\gamma \end{pmatrix} \]\[ = (a+a_2)+(b+b_2)i+(c+c_2)j+(d+d_2)k = \begin{pmatrix} a + a_2 + \iota(b + b_2) & -(c + c_2 + \iota(d + d_2))\\ c + c_2 - \iota(d + d_2) & a + a_2 -\iota(b + \iota b_2) \end{pmatrix} = \begin{pmatrix} \alpha + \gamma & -(\beta + \delta)\\ \bar\beta + \bar\delta & \bar\alpha + \bar\gamma \end{pmatrix} = \begin{pmatrix} \alpha + \gamma & -(\beta + \delta)\\ \overline{\beta + \delta} & \overline{\alpha + \gamma} \end{pmatrix} \]
\[ A - A_2 = a+bi+cj+dk - (a_2+b_2i+c_2j+d_2k) = \begin{pmatrix} a + \iota b & -(c + \iota d)\\ c - \iota d & a - \iota b \end{pmatrix} - \begin{pmatrix} a_2 + \iota b_2 & -(c_2 + \iota d_2)\\ c_2 - \iota d_2 & a_2 - \iota b_2 \end{pmatrix} = \begin{pmatrix} \alpha & -\beta\\ \bar\beta & \bar\alpha \end{pmatrix} - \begin{pmatrix} \gamma & -\delta\\ \bar\delta & \bar\gamma \end{pmatrix} \]\[ = (a-a_2)+(b-b_2)i+(c-c_2)j+(d-d_2)k = \begin{pmatrix} a - a_2 + \iota(b - b_2) & -(c - c_2 + \iota(d - d_2))\\ c - c_2 - \iota(d - d_2) & a - a_2 - \iota(b - b_2) \end{pmatrix} = \begin{pmatrix} \alpha - \gamma & -(\beta - \delta)\\ \bar\beta - \bar\delta & \bar\alpha - \bar\gamma \end{pmatrix} = \begin{pmatrix} \alpha - \gamma & -(\beta - \delta)\\ \overline{\beta - \delta} & \overline{\alpha - \gamma} \end{pmatrix} \]
\[ AA_2 = (a+bi+cj+dk)(a_2+b_2i+c_2j+d_2k) = \begin{pmatrix} a + \iota b & -(c + \iota d)\\ c - \iota d & a - \iota b \end{pmatrix} \begin{pmatrix} a_2 + \iota b_2 & -(c_2 + \iota d_2)\\ c_2 - \iota d_2 & a_2 - \iota b_2 \end{pmatrix} = \begin{pmatrix} \alpha & -\beta\\ \bar\beta & \bar\alpha \end{pmatrix} \begin{pmatrix} \gamma & -\delta\\ \bar\delta & \bar\gamma \end{pmatrix} \]\[ =aa_2-bb_2-cc_2-dd_2 +(ab_2+ba_2+cd_2-dc_2)i +(ac_2-bd_2+ca_2+db_2)j +(ad_2+bc_2-cb_2+da_2)k \]\[ = \begin{pmatrix} (a + \iota b)(a_2 + \iota b_2) - (c + \iota d)(c_2 - \iota d_2) & -(a + \iota b)(c_2 + \iota d_2) - (c + \iota d)(a_2 - \iota b_2)\\ (c - \iota d)(a_2 + \iota b_2) + (a - \iota b)(c_2 - \iota d_2) & -(c - \iota d)(c_2 + \iota d_2) + (a - \iota b)(a_2 - \iota b_2) \end{pmatrix} \]\[ = \begin{pmatrix} aa_2 - bb_2 - cc_2 - dd_2 + \iota(ab_2 + ba_2 + cd_2 - dc_2) & -(ac_2 - bd_2 + ca_2 + db_2 + \iota(ad_2 + bc_2 - cb_2 + da_2))\\ ac_2 - bd_2 + ca_2 + db_2 - \iota(ad_2 + bc_2 - cb_2 + da_2) & aa_2 - bb_2 - cc_2 - dd_2 - \iota(ab_2 + ba_2 + cd_2 - dc_2) \end{pmatrix} \]\[ = \begin{pmatrix} \alpha\gamma - \beta\bar\delta & -(\alpha\delta + \beta\bar\gamma)\\ \bar\alpha\bar\delta + \bar\beta\gamma & \bar\alpha\bar\gamma - \bar\beta\delta \end{pmatrix} = \begin{pmatrix} \alpha\gamma - \beta\bar\delta & -(\alpha\delta + \beta\bar\gamma)\\ \overline{\alpha\delta + \beta\bar\gamma} & \overline{\alpha\gamma - \beta\bar\delta} \end{pmatrix} \]
\[ A\bar{A} = (a+bi+cj+dk)(a-bi-cj-dk) = \begin{pmatrix} \alpha & -\beta\\ \bar\beta & \bar\alpha \end{pmatrix} \begin{pmatrix} \bar\alpha & \beta\\ -\bar\beta & \alpha \end{pmatrix} = \begin{pmatrix} \alpha\bar\alpha + \beta\bar\beta & 0\\ 0 & \alpha\bar\alpha + \beta\bar\beta \end{pmatrix} = \begin{pmatrix} a^2 + b^2 + c^2 + d^2 & 0\\ 0 & a^2 + b^2 + c^2 + d^2 \end{pmatrix} \]\[ =aa-b(-b)-c(-c)-d(-d) +(a(-b)+ba+c(-d)-d(-c))i +(a(-c)-b(-d)+ca+d(-b))j +(a(-d)+b(-c)-c(-b)+da)k =a^2 + b^2 + c^2 + d^2 \]\[ = \begin{pmatrix} aa - b(-b) - c(-c) - d(-d) + \iota(a(-b) + ba + c(-d) - d(-c)) & -(a(-c) - b(-d) + ca + d(-b) + \iota(a(-d) + b(-c) - c(-b) + da))\\ a(-c) - b(-d) + ca + d(-b) - \iota(a(-d) + b(-c) - c(-b) + da) & aa - b(-b) - c(-c) - d(-d) - \iota(a(-b) + ba + c(-d) - d(-c)) \end{pmatrix} \]\[ = (a^2 + b^2 + c^2 + d^2) \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \]
where \(A\neq 0\), \(\left|A\right|^2\neq 0\)\[ A^{-1} = \frac{\bar{A}}{\left|A\right|^2} = \frac{\alpha - i\bar\beta}{\left|A\right|^2} = \frac1{\left|A\right|^2} \begin{pmatrix} \bar\alpha & \beta\\ -\bar\beta & \alpha \end{pmatrix} \]
\[ e^{a+bi+cj+dk} = \sum_{\forall n\in\mathbb{N}_0}\frac{(a+bi+cj+dk)^n}{n!} = e^a\left(\cos\left|bi+cj+dk\right| + \frac{bi+cj+dk}{\left|bi+cj+dk\right|}\sin\left|bi+cj+dk\right|\right) \]
\[ A = \begin{pmatrix} \alpha & \beta\\ \epsilon & \kappa \end{pmatrix} ,\quad A^T = \begin{pmatrix} \alpha & \epsilon\\ \beta & \kappa \end{pmatrix} ,\quad \bar{A} = \begin{pmatrix} \bar\alpha & \bar\beta\\ \bar\epsilon & \bar\kappa \end{pmatrix} ,\quad A^* = \overline{A^T} = \bar{A}^T = \begin{pmatrix} \bar\alpha & \bar\epsilon\\ \bar\beta & \bar\kappa \end{pmatrix} ,\quad (A^*)^* = \begin{pmatrix} \bar\alpha & \bar\epsilon\\ \bar\beta & \bar\kappa \end{pmatrix}^* = \begin{pmatrix} \alpha & \beta\\ \epsilon & \kappa \end{pmatrix} = A ,\quad \alpha, \beta, \gamma, \delta \in \mathbb{C} \] \[ \bar{A}\bar{B} = \begin{pmatrix} \bar\alpha & \bar\beta\\ \bar\epsilon & \bar\kappa \end{pmatrix} \begin{pmatrix} \bar\gamma & \bar\delta\\ \bar\lambda & \bar\mu \end{pmatrix} = \begin{pmatrix} \bar\alpha\bar\gamma + \bar\beta\bar\lambda & \bar\alpha\bar\delta + \bar\beta\bar\mu\\ \bar\epsilon\bar\gamma + \bar\kappa\bar\lambda & \bar\epsilon\bar\delta + \bar\kappa\bar\mu \end{pmatrix} = \overline{AB} \] \[ (AB)^T = \left( \begin{pmatrix} \alpha & \beta\\ \epsilon & \kappa \end{pmatrix} \begin{pmatrix} \gamma & \delta\\ \lambda & \mu \end{pmatrix} \right)^T = \begin{pmatrix} \alpha\gamma + \beta\lambda & \alpha\delta + \beta\mu\\ \epsilon\gamma + \kappa\lambda & \epsilon\delta + \kappa\mu \end{pmatrix}^T = \begin{pmatrix} \alpha\gamma + \beta\lambda & \epsilon\gamma + \kappa\lambda\\ \alpha\delta + \beta\mu & \epsilon\delta + \kappa\mu \end{pmatrix} \] \[ B^TA^T = \begin{pmatrix} \gamma & \delta\\ \lambda & \mu \end{pmatrix}^T \begin{pmatrix} \alpha & \beta\\ \epsilon & \kappa \end{pmatrix}^T = \begin{pmatrix} \gamma & \lambda\\ \delta & \mu \end{pmatrix} \begin{pmatrix} \alpha & \epsilon\\ \beta & \kappa \end{pmatrix} = \begin{pmatrix} \alpha\gamma + \beta\lambda & \epsilon\gamma + \kappa\lambda\\ \alpha\delta + \beta\mu & \epsilon\delta + \kappa\mu \end{pmatrix} = (AB)^T \] \[ (AB)^* = \left( \begin{pmatrix} \alpha & \beta\\ \epsilon & \kappa \end{pmatrix} \begin{pmatrix} \gamma & \delta\\ \lambda & \mu \end{pmatrix} \right)^* = \begin{pmatrix} \alpha\gamma + \beta\lambda & \alpha\delta + \beta\mu\\ \epsilon\gamma + \kappa\lambda & \epsilon\delta + \kappa\mu \end{pmatrix}^* = \begin{pmatrix} \bar\alpha\bar\gamma + \bar\beta\bar\lambda & \bar\epsilon\bar\gamma + \bar\kappa\bar\lambda\\ \bar\alpha\bar\delta + \bar\beta\bar\mu & \bar\epsilon\bar\delta + \bar\kappa\bar\mu \end{pmatrix} \] \[ B^*A^* = \begin{pmatrix} \gamma & \delta\\ \lambda & \mu \end{pmatrix}^* \begin{pmatrix} \alpha & \beta\\ \epsilon & \kappa \end{pmatrix}^* = \begin{pmatrix} \bar\gamma & \bar\lambda\\ \bar\delta & \bar\mu \end{pmatrix} \begin{pmatrix} \bar\alpha & \bar\epsilon\\ \bar\beta & \bar\kappa \end{pmatrix} = \begin{pmatrix} \bar\alpha\bar\gamma + \bar\beta\bar\lambda & \bar\epsilon\bar\gamma + \bar\kappa\bar\lambda\\ \bar\alpha\bar\delta + \bar\beta\bar\mu & \bar\epsilon\bar\delta + \bar\kappa\bar\mu \end{pmatrix} = (AB)^* \] \[ n\text{-order Hermite matrix: } H = H^* ,\qquad\text{where }H\in P_n \] \[ n\text{-order Unitary matrix: } UU^* = 1_n ,\qquad\text{where }U\in U_n \] \[ e^X = \sum_{\forall n\in\mathbb{N}_0}\frac1{n!}X^n ,\qquad\text{where }X\in M_n(\mathbb{C}),\quad e^X\in GL_n(\mathbb{C}) \] \[ e^H = \sum_{\forall n\in\mathbb{N}_0}\frac1{n!}H^n ,\qquad\text{where }H\in P_n,\quad e^H\in P_n \]