Ho̍k-kiōng-tam

Tī sò͘-ha̍k tiong, chi̍t ê ho̍k-cha̍p-sò͘ $z$ ê ho̍k-kiōng-tam (Hàn-jī: 複共擔, Eng-gí: complex conjugate) ${\overline {z}}$ mā sī chi̍t ê ho̍k-cha̍p-sò͘, ${\overline {z}}$ ê si̍t-pó͘ kah $z$ sio-kâng, ${\overline {z}}$ hi-pō͘ kah $z$ pêⁿ-tōa tān-sī hû-hō tùi-hoán ê sò͘. Khiā-sǹg-kóng, nā-chún $a$  kap $b$ sī si̍t-sò͘, $z=a+bi$ ê ho̍k-kiōng-tam tiō sī ${\overline {z}}=a-bi$. Pí-lūn-kóng,

${\textstyle {\overline {-4+3i}}=-4-3i}$

${\textstyle {\overline {8}}=8}$ (si̍t-sò͘ ê ho̍k-kiōng-tam sī i ka-kī)

${\textstyle {\overline {2i}}=-2i}$ (sûn hi-sò͘ ê ho̍k-kiōng-tam sī i ê tùi-hoán sò͘)

Lán nā iōng-tio̍h ke̍k-chò-piau piáu-sī, ē-sái án-ne tēng-gī ho̍k-kiōng-tam:

${\overline {re^{i\theta }}}=re^{-i\theta }$

kî-tiong in ê hok-kak hû-hō tùi-hoán. Che ē-sái iōng Euler kong-sek lâi chèng-bêng.

Miâ-chheng

Eng-gí conjugate goân-chū Latin-gí ê coniugāre (“liân chò-hóe, gû-taⁿ”), iû con- (“chò-hóe”) +‎ iugāre (“ chiap, pa̍k, liân”) kò͘-sêng.

Tiong-kok-ōe kā e̍k-chò 共軛. Ji̍t-pún-ōe e̍k-chò 共役. Tâi-oân-ōe e̍k-chò 共擔.

Kì-hō

Ho̍k-kiōng-tam ê kì-hō ū nn̄g chióng: vinculum ${\overline {z}}$ kap chheⁿ-tiám $z^{*}$.

Tē 1 chióng sò͘-ha̍k-ka khah ài iōng.

Tē 2 chióng bu̍t-lí-ha̍k, tiān-ki kang-têng, tiān-náu kho-ha̍k téng-téng léng-he̍k khah ài iōng.

It-poaⁿ-hòa

Chi̍t ê ho̍k hâng-lia̍t ê ho̍k-kiōng-tam tiō sī ta̍k-ê iàu-sò͘ lóng ka i piàn-chò ho̍k-kiōng-tam. Pí-lūn-kóng,

$\mathbf {A} ={\begin{pmatrix}2+3i&{\sqrt {3}}i\\1-0.5i&-6\end{pmatrix}},\quad {\overline {\mathbf {A} }}={\begin{pmatrix}2-3i&-{\sqrt {3}}i\\1+0.5i&-6\end{pmatrix}}.$

Sèng-chit

Tùi jīm-hô nn̄g ê ho̍k-cha̍p-sò͘ $z$ kap $w$, lán ū

• ${\overline {\left({\overline {z}}\right)}}=z$.
• ${\overline {z+w}}={\overline {z}}+{\overline {w}}$.
• ${\overline {z-w}}={\overline {z}}-{\overline {w}}$.
• ${\overline {zw}}={\overline {z}}\cdot {\overline {w}}$.
• ${\overline {\left({\frac {z}{w}}\right)}}={\frac {\overline {z}}{\overline {w}}}$, chún-kóng $w\neq 0$.
• $z{\overline {z}}={\left|z\right|}^{2}$.
• $\left\vert {\overline {z}}\right\vert =\left\vert z\right\vert$.
• $z^{-1}={\frac {\overline {z}}{{\left|z\right|}^{2}}}$, chún-kóng $z\neq 0$.
• ${\overline {z^{n}}}=\left({\overline {z}}\right)^{n}$, tùi só͘-ū ê $n\in \mathbb {Z}$
• $\exp \left({\overline {z}}\right)={\overline {\exp(z)}}$
• $\ln \left({\overline {z}}\right)={\overline {\ln(z)}}$, chún-kóng $z\neq 0$.

See also

• Hermite kiōng-tam