Ho̍k-kiōng-tam

佇 (Tī)數學 (sò͘-ha̍k)中 (tiong)，一 (chi̍t)个 (ê)複雜數 (ho̍k-cha̍p-sò͘) 解析失敗 (Chīn-liōng iōng MathML (chhì-giām-sèng--ê)：從伺服器 "https://wikimedia.org/api/rest_v1/" 收到無效的回應 ("Math extension cannot connect to Restbase.")。): {\displaystyle z} 的 (ê)複共擔 (ho̍k-kiōng-tam) (漢字: 複共擔，英語: complex conjugate) 解析失敗 (Chīn-liōng iōng MathML (chhì-giām-sèng--ê)：從伺服器 "https://wikimedia.org/api/rest_v1/" 收到無效的回應 ("Math extension cannot connect to Restbase.")。): {\displaystyle \overline{z}} 嘛 (mā)是 (sī)一个複雜數，解析失敗 (Chīn-liōng iōng MathML (chhì-giām-sèng--ê)：從伺服器 "https://wikimedia.org/api/rest_v1/" 收到無效的回應 ("Math extension cannot connect to Restbase.")。): {\displaystyle \overline{z}} 的實補 (si̍t-pó͘)佮 (kah) 解析失敗 (Chīn-liōng iōng MathML (chhì-giām-sèng--ê)：從伺服器 "https://wikimedia.org/api/rest_v1/" 收到無效的回應 ("Math extension cannot connect to Restbase.")。): {\displaystyle z} 相仝 (sio-kâng)，解析失敗 (Chīn-liōng iōng MathML (chhì-giām-sèng--ê)：從伺服器 "https://wikimedia.org/api/rest_v1/" 收到無效的回應 ("Math extension cannot connect to Restbase.")。): {\displaystyle \overline{z}} 虛部 (hi-pō͘)佮 解析失敗 (Chīn-liōng iōng MathML (chhì-giām-sèng--ê)：從伺服器 "https://wikimedia.org/api/rest_v1/" 收到無效的回應 ("Math extension cannot connect to Restbase.")。): {\displaystyle z} 平大 (pêⁿ-tōa)但是 (tān-sī)符號 (hû-hō)對反 (tùi-hoán)的數 (sò͘)。徛算講 (Khiā-sǹg-kóng)，若準 (nā-chún) 解析失敗 (Chīn-liōng iōng MathML (chhì-giām-sèng--ê)：從伺服器 "https://wikimedia.org/api/rest_v1/" 收到無效的回應 ("Math extension cannot connect to Restbase.")。): {\displaystyle a} 佮 (kap) 解析失敗 (Chīn-liōng iōng MathML (chhì-giām-sèng--ê)：從伺服器 "https://wikimedia.org/api/rest_v1/" 收到無效的回應 ("Math extension cannot connect to Restbase.")。): {\displaystyle b} 是實數 (si̍t-sò͘)，解析失敗 (Chīn-liōng iōng MathML (chhì-giām-sèng--ê)：從伺服器 "https://wikimedia.org/api/rest_v1/" 收到無效的回應 ("Math extension cannot connect to Restbase.")。): {\displaystyle z=a+bi} 的複共擔就 (tiō)是 解析失敗 (Chīn-liōng iōng MathML (chhì-giām-sèng--ê)：從伺服器 "https://wikimedia.org/api/rest_v1/" 收到無效的回應 ("Math extension cannot connect to Restbase.")。): {\displaystyle \overline{z}=a-bi} 。比論講 (Pí-lūn-kóng)，

解析失敗 (Chīn-liōng iōng MathML (chhì-giām-sèng--ê)：從伺服器 "https://wikimedia.org/api/rest_v1/" 收到無效的回應 ("Math extension cannot connect to Restbase.")。): {\textstyle \overline{-4+3i} = -4-3i}

解析失敗 (Chīn-liōng iōng MathML (chhì-giām-sèng--ê)：從伺服器 "https://wikimedia.org/api/rest_v1/" 收到無效的回應 ("Math extension cannot connect to Restbase.")。): {\textstyle \overline{8} = 8} (實數的複共擔是伊 (i)家己 (ka-kī))

解析失敗 (Chīn-liōng iōng MathML (chhì-giām-sèng--ê)：從伺服器 "https://wikimedia.org/api/rest_v1/" 收到無效的回應 ("Math extension cannot connect to Restbase.")。): {\textstyle \overline{2i} = -2i} (純 (sûn)虛數 (hi-sò͘)的複共擔是伊的對反數)

咱 (Lán)若 (nā)用著 (iōng-tio̍h)極做標 (ke̍k-chò-piau)表示 (piáu-sī)，會使 (ē-sái)按呢 (án-ne)定義 (tēng-gī)複共擔:

解析失敗 (Chīn-liōng iōng MathML (chhì-giām-sèng--ê)：從伺服器 "https://wikimedia.org/api/rest_v1/" 收到無效的回應 ("Math extension cannot connect to Restbase.")。): {\displaystyle \overline{re^{i \theta}} = re^{-i \theta}}

其中 (kî-tiong)怹 (in)的福角 (hok-kak)符號對反。這 (Che)會使用 (iōng)Euler公式 (kong-sek)來 (lâi)證明 (chèng-bêng)。

名稱 (Miâ-chheng)

英語 (Eng-gí)conjugate源自 (goân-chū)Latin語 (gí)的coniugāre (“聯 (liân)做伙 (chò-hóe)，牛今 (gû-taⁿ)”)，由 (iû)con- (“做夥 (chò-hóe)”) +‎ iugāre (“ 接 (chiap)，縛 (pa̍k)，連 (liân)”) 構成 (kò͘-sêng)。

中國畫 (Tiong-kok-ōe)共 (kā)役做 (e̍k-chò)共軛。日本畫 (Ji̍t-pún-ōe)譯做 (e̍k-chò)共役。台灣話 (Tâi-oân-ōe)譯做共擔。

記號 (Kì-hō)

複共擔的記號有 (ū)兩 (nn̄g)種 (chióng): vinculum 解析失敗 (Chīn-liōng iōng MathML (chhì-giām-sèng--ê)：從伺服器 "https://wikimedia.org/api/rest_v1/" 收到無效的回應 ("Math extension cannot connect to Restbase.")。): {\displaystyle \overline{z}} 佮星點 (chheⁿ-tiám) 解析失敗 (Chīn-liōng iōng MathML (chhì-giām-sèng--ê)：從伺服器 "https://wikimedia.org/api/rest_v1/" 收到無效的回應 ("Math extension cannot connect to Restbase.")。): {\displaystyle z^*} 。

第 (Tē)1種數學家 (sò͘-ha̍k-ka)較 (khah)愛 (ài)用。

第2種物理學 (bu̍t-lí-ha̍k)，電機 (tiān-ki)工程 (kang-têng)，電腦 (tiān-náu)科學 (kho-ha̍k)等等 (téng-téng)領域 (léng-he̍k)較愛用。

一般化 (It-poaⁿ-hòa)

一个服 (ho̍k)行列 (hâng-lia̍t)的複共擔就是逐个 (ta̍k-ê)要數 (iàu-sò͘)攏 (lóng)加 (ka)伊變做 (piàn-chò)複共擔。比論講，

解析失敗 (Chīn-liōng iōng MathML (chhì-giām-sèng--ê)：從伺服器 "https://wikimedia.org/api/rest_v1/" 收到無效的回應 ("Math extension cannot connect to Restbase.")。): {\displaystyle \mathbf{A} = \begin{pmatrix} 2+3i & \sqrt{3} 伊 \\ 1-0.5i & -6\end{pmatrix} ,\quad \overline{\mathbf{A}} = \begin{pmatrix} 2-3i & -\sqrt{3} 伊 \\ 1+0.5i & -6\end{pmatrix}.}

性質 (Sèng-chit)

對 (Tùi)任何 (jīm-hô)兩个複雜數 解析失敗 (Chīn-liōng iōng MathML (chhì-giām-sèng--ê)：從伺服器 "https://wikimedia.org/api/rest_v1/" 收到無效的回應 ("Math extension cannot connect to Restbase.")。): {\displaystyle z} 佮 解析失敗 (Chīn-liōng iōng MathML (chhì-giām-sèng--ê)：從伺服器 "https://wikimedia.org/api/rest_v1/" 收到無效的回應 ("Math extension cannot connect to Restbase.")。): {\displaystyle w} ，咱有

• 解析失敗 (Chīn-liōng iōng MathML (chhì-giām-sèng--ê)：從伺服器 "https://wikimedia.org/api/rest_v1/" 收到無效的回應 ("Math extension cannot connect to Restbase.")。): {\displaystyle \overline{\left(\overline{z}\right)}=z} 。
• 解析失敗 (Chīn-liōng iōng MathML (chhì-giām-sèng--ê)：從伺服器 "https://wikimedia.org/api/rest_v1/" 收到無效的回應 ("Math extension cannot connect to Restbase.")。): {\displaystyle \overline{z + w} = \overline{z} + \overline{w}} 。
• 解析失敗 (Chīn-liōng iōng MathML (chhì-giām-sèng--ê)：從伺服器 "https://wikimedia.org/api/rest_v1/" 收到無效的回應 ("Math extension cannot connect to Restbase.")。): {\displaystyle \overline{z - w} = \overline{z} - \overline{w}} 。
• 解析失敗 (Chīn-liōng iōng MathML (chhì-giām-sèng--ê)：從伺服器 "https://wikimedia.org/api/rest_v1/" 收到無效的回應 ("Math extension cannot connect to Restbase.")。): {\displaystyle \overline{zw} = \overline{z}\cdot \overline{w}} 。
• 解析失敗 (Chīn-liōng iōng MathML (chhì-giām-sèng--ê)：從伺服器 "https://wikimedia.org/api/rest_v1/" 收到無效的回應 ("Math extension cannot connect to Restbase.")。): {\displaystyle \overline{\left(\frac{z}{w}\right)} = \frac{\overline{z}}{\overline{w}}} ，準講 (chún-kóng) 解析失敗 (Chīn-liōng iōng MathML (chhì-giām-sèng--ê)：從伺服器 "https://wikimedia.org/api/rest_v1/" 收到無效的回應 ("Math extension cannot connect to Restbase.")。): {\displaystyle w \neq 0} 。
• 解析失敗 (Chīn-liōng iōng MathML (chhì-giām-sèng--ê)：從伺服器 "https://wikimedia.org/api/rest_v1/" 收到無效的回應 ("Math extension cannot connect to Restbase.")。): {\displaystyle z\overline{z} = {\left| z \right|}^2} 。
• 解析失敗 (Chīn-liōng iōng MathML (chhì-giām-sèng--ê)：從伺服器 "https://wikimedia.org/api/rest_v1/" 收到無效的回應 ("Math extension cannot connect to Restbase.")。): {\displaystyle \left\vert \overline{z} \right\vert = \left\vert z \right\vert} 。
• 解析失敗 (Chīn-liōng iōng MathML (chhì-giām-sèng--ê)：從伺服器 "https://wikimedia.org/api/rest_v1/" 收到無效的回應 ("Math extension cannot connect to Restbase.")。): {\displaystyle z^{-1} = \frac{\overline{z}}{{\left| z \right|}^2}} ，準講 解析失敗 (Chīn-liōng iōng MathML (chhì-giām-sèng--ê)：從伺服器 "https://wikimedia.org/api/rest_v1/" 收到無效的回應 ("Math extension cannot connect to Restbase.")。): {\displaystyle z \neq 0} 。
• 解析失敗 (Chīn-liōng iōng MathML (chhì-giām-sèng--ê)：從伺服器 "https://wikimedia.org/api/rest_v1/" 收到無效的回應 ("Math extension cannot connect to Restbase.")。): {\displaystyle \overline{z^n} = \left(\overline{z}\right)^n} ，對所有的 解析失敗 (Chīn-liōng iōng MathML (chhì-giām-sèng--ê)：從伺服器 "https://wikimedia.org/api/rest_v1/" 收到無效的回應 ("Math extension cannot connect to Restbase.")。): {\displaystyle n \in \Z}
• 解析失敗 (Chīn-liōng iōng MathML (chhì-giām-sèng--ê)：從伺服器 "https://wikimedia.org/api/rest_v1/" 收到無效的回應 ("Math extension cannot connect to Restbase.")。): {\displaystyle \exp\left(\overline{z}\right) = \overline{\exp(z)}}
• 解析失敗 (Chīn-liōng iōng MathML (chhì-giām-sèng--ê)：從伺服器 "https://wikimedia.org/api/rest_v1/" 收到無效的回應 ("Math extension cannot connect to Restbase.")。): {\displaystyle \ln\left(\overline{z}\right) = \overline{\ln(z)}} ，準講 解析失敗 (Chīn-liōng iōng MathML (chhì-giām-sèng--ê)：從伺服器 "https://wikimedia.org/api/rest_v1/" 收到無效的回應 ("Math extension cannot connect to Restbase.")。): {\displaystyle z \neq 0} 。

See also

• Hermite共擔