Tlhahlobo e Rarahaneng ea Lipalo
Tlhahlobo e rarahaneng ke lekala la lipalo le ithutang mesebetsi e nang le linomoro tse rarahaneng le thepa ea tsona. Linomoro tse rarahaneng, hangata li ngotsoe ka mokhoa oa \( z = a + bi \) moo \(a\) le \(b\) e leng linomoro tsa 'nete le \(i^2 = -1\), qalong li hlahile e le sesebelisoa sa ho rarolla li-quadratic equation tse se nang metso ea 'nete. Leha ho le joalo, ha lipalo li ntse li tsoela pele, linomoro tse rarahaneng li fetohile motheo oa bohlokoa bakeng sa likhopolo-taba tse ngata tsa sejoale-joale—ho tloha tlhahlobong le fisiks ea lipalo ho ea boenjiniere le ts'ebetsong ea matšoao. Ho ikhetha ha tlhahlobo e rarahaneng ho lutse 'neteng ea hore mesebetsi e rarahaneng e "ntle" (ea tlhahlobo) e na le sebopeho se matla haholo, hoo liphetho tse ngata tse thata tlhahlobong ea 'nete li fetohang tse ntle lefatšeng le rarahaneng.
1. Linomoro tse rarahaneng le litlhaloso tsa tsona
Linomoro tse rarahaneng li ka emeloa ka mefuta e mengata. Sebopeho se tloaelehileng haholo ke sebopeho sa Cartesian \(z = x + iy\), moo \(x\) e leng karolo ea 'nete 'me \(y\) e le karolo e inahaneloang. Ho latela jeometri, linomoro tse rarahaneng li ka bonoa e le lintlha tse ka har'a sefofane se rarahaneng (kapa sefofane sa Argand), ka mothapo o otlolohileng bakeng sa karolo ea 'nete le mothapo o otlolohileng bakeng sa karolo e inahaneloang.
Ntle le sebopeho sa Cartesian, ho na le sebopeho se thusang haholo sa polar:
\[
z = r(\cos\theta + i\sin\theta),
\]
ka \(r = |z| = \sqrt{x^2+y^2}\) e le modulus le \(\theta\) e le ngangisano (sekhutlo se nang le mothapo wa nnete). Sebopeho sena hangata se ngolwa ka bokgutshwanyane ho sebediswa foromo ya Euler:
\[
e^{i\theta} = \cos\theta + i\sin\theta,
\]
sehinga
\[
z = re^{i\theta}.
\]
Sebopeho sa polar se nolofatsa katiso, karohano, le tlhaloso e rarahaneng. Mohlala, haeba \(z_1 = r_1e^{i\theta_1}\) le \(z_2 = r_2e^{i\theta_2}\), joale \(z_1z_2 = r_1r_2e^{i(\theta_1+\theta_2)}\). Sena se hlalosa hore na ke hobane'ng ha ho potoloha le ho lekanya ho ka etsoa mohlala ka tlhaho ho sebelisoa linomoro tse rarahaneng.
2. Mesebetsi e rarahaneng le mehopolo ea tlhahlobo
Tlhahlobong ea 'nete, mesebetsi e ka khetholloang ha se kamehla e leng "e tloaelehileng ka ho phethahetseng." Ka lehlakoreng le leng, tlhahlobong e rarahaneng, litlhoko tsa ho khetholla li thata haholo. Mosebetsi o rarahaneng \(f(z)\) o bitsoa mokhoa o rarahaneng oa ho khetholla ntlheng \(z_0\) haeba moeli
\[
f'(z_0)=\lim_{z\to z_0}\frac{f(z)-f(z_0)}{z-z_0}
\]
e teng 'me e na le boleng bo tšoanang bakeng sa litsela tsohle tsa ho atamela \(z \to z_0\). Kaha \(z\) e sebakeng sa mahlakore a mabeli (seo ho nahanoang ka sona), moeli ona o na le lithibelo tse ngata ho feta phapang moleng oa 'nete.
Haeba mosebetsi o rarahaneng o ka kgetholoha sebakeng se bulehileng, o bitswa holomorphic. Haeba o le holomorphic sebakeng sohle se rarahaneng, o bitswa e felletseng (e^z, sin z, le polynomials, mohlala). Sephetho se le seng se makatsang: mosebetsi wa holomorphic o ka atoloswa ka bohona e le letoto la matla (letoto la Taylor) ho potoloha ntlha sebakeng sa ona. Sena se bolela hore "ho kgetholoha ho rarahaneng" ha e le hantle ho lekana le "analytic" ka kutlwisiso ya letoto la matla - ketsahalo e matla haholo.
3. Tekanyo ea Cauchy-Riemann
Ho utloisisa hore na ke hobane'ng ha phapang e rarahaneng e le matla hakaale, re ka arola mosebetsi o rarahaneng ka likarolo tsa 'nete le tse inahaneloang.
\[
f(z)=u(x,y)+iv(x,y), \quad z=x+iy.
\]
Boemo bo hlokahalang (mme maemong a mangata bo boetse bo lekane) bakeng sa hore \(f\) e be holomorphic ke hore di-equation tsa Cauchy-Riemann di kgotsofatswe:
\[
\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}, \qquad \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}.
\]
Tekanyo ena e amana ka thata le di-derivative tse sa fellang tsa \(u\) le \(v\). Ditlamorao tsa yona di tebile: haeba \(u\) le \(v\) di ka kgetholoha hantle mme di kgotsofatsa thepa ya Cauchy-Riemann, jwale \(f\) e hlophisitswe hantle haholo. Ha e le hantle, \(u\) le \(v\) hangata di amana le mehopolo ya mmele jwalo ka bokgoni le phallo (mohlala, ka di-dynamics tse ntle tsa mokelikeli), kaha ka bobedi e ka ba mesebetsi ya harmonic.
4. Mekhahlelo e rarahaneng le likhopolo-taba tsa motheo
Tlhahlobo e rarahaneng e boetse e tumme ka khopolo-taba ea eona e ntle ea li-integral. Karolo e kopaneng tseleng (contour) e hlalosoa bakeng sa ts'ebetso \(f(z)\) e kobehileng \(\gamma\) sebakeng se rarahaneng:
\[
\int_\gamma f(z)\,dz.
\]
Ho fapana le di-integral tsa nnete ka di-interval, di-integral tse rarahaneng di kenyelletsa ditsela tse ka kobehang, tsa potoloha, kapa tsa etsa di-loop.
E 'ngoe ea litšiea tse kholo ke Cauchy Integral Theorem: haeba \(f\) e le holomorphic sebakeng se "ntle" 'me \(\gamma\) e le kobeho e koetsoeng sebakeng seo, joale
\[
\int_\gamma f(z)\,dz = 0.
\]
Tšusumetso ea eona e bile kholo haholo: ho tsoa khopolo-taba ena, ho hlahile liphetho tse ling tse ngata tsa bohlokoa, ho kenyeletsoa le Foromo ea Cauchy Integral e bolelang hore boleng ba mosebetsi oa holomorphic ntlheng e itseng bo ka baloa feela ho tsoa boleng ba mosebetsi moeling oa sebaka:
\[
f(z_0)=\frac{1}{2\pi i}\int_\gamma \frac{f(z)}{z-z_0}\,dz.
\]
Foromo ena e tiisa hore mosebetsi oa holomorphic o "ikemiselitsoe" ke boitšoaro ba oona moeling. E boetse e fana ka 'nete ea hore li-derivative tsohle tsa \(f^{(n)}(z_0)\) li ka hlalosoa e le li-integral, ka hona ho netefatsa boemo bo phahameng haholo ba ho ba le botsitso bakeng sa mesebetsi ea holomorphic.
5. Letoto la Laurent le masalla
Hase mesebetsi yohle e rarahaneng e boreleli hodima domain yohle. Mesebetsi e mengata e na le dintlha tse le nngwe (mohlala, \(1/z\) ho \(z=0\)). Ho sekaseka boitshwaro bo potileng bong bo le bong, letoto la Laurent le sebediswa:
\[
f(z) = \sum_{n=-\infty}^{\infty} a_n (z-z_0)^n.
\]
Ho fapana le letoto la Taylor le nang le di-exponents tse seng mpe feela, letoto la Laurent le dumella di-exponents tse mpe tse emelang karolo e le nngwe.
Coefficient \(a_{-1}\) letotong la Laurent e bohlokoa haholo 'me e bitsoa masala. Ho tsoa khopolong ea masala ho hlaha Residue Theorem, e bolelang hore motsoako oa mosebetsi o tsamaeang le mothapo o koetsoeng o ka baloa ho tsoa kakaretsong ea masala a singularities ka har'a mothapo:
\[
\int_\gamma f(z)\,dz = 2\pi i \sum \text{Res}(f, z_k).
\]
Teori ena e fana ka sesebelisoa se matla sa ho bala likarolo tse thata tsa 'nete, haholo-holo tse kenyeletsang mesebetsi ea kelello le ea trigonometric, ka ho li fetola mathata a rarahaneng.
6. Litšebeliso tsa 'mapa o lumellanang le oa jeometri
Mesebetsi ea Holomorphic (e nang le li-derivative tse seng tsa zero) e lumellana, ke hore, e boloka likhutlo sebakeng sa heno. Sena se bolela hore mesebetsi e joalo e hokahanya li-lattice tse nyane tse sebakeng se rarahaneng le li-lattice tse ling ntle le ho fetola likhutlo tseo li kopanang le tsona. Limmapa tse lumellanang li na le thuso haholo ho jiometri le fisiks, mohlala, ho rarolla mathata a matla a motlakase kapa phallo ea mokelikeli ka ho fetola libaka tse rarahaneng hore e be tse bonolo (joalo ka ho etsa 'mapa oa sebaka se nang le moeli o kobehileng ho selikalikoe).
Mohlala o motle ke phetoho ea Möbius:
\[
f(z)=\frac{az+b}{cz+d},
\]
ka \(ad-bc\neq 0\). Phetoho ena e kopanya mela le dikotwana ho ya ho mela kapa dikotwana, mme e bapala karolo e kgolo thutong e rarahaneng ya jeometri le thuto ya tshebetso.
7. Kamano le masimo a mang
Tlhahlobo e rarahaneng ha se khopolo-taba feela e sa utloahaleng. E arola mafapheng a mangata:
1. Fisiks: quantum mechanics, khopolo-taba ea tšimo, le electrodynamics li sebelisa mesebetsi e rarahaneng haholo le liphetoho tsa Fourier/Laplace tse amanang haufi-ufi le tlhahlobo e rarahaneng.
2. Mokhoa: Tlhahlobo ea potoloho ea motlakase (impedance) hangata e ngoloa ka linomoro tse rarahaneng; taolo ea sistimi e sebelisa liphetoho tsa Laplace tse nang le litlhaloso tse rarahaneng.
3. Ts'ebetso ea matšoao: frequency spectrum le Fourier transform li amana haufi-ufi le exponentials e rarahaneng.
4. Lipalo tse hloekileng: khopolo-taba ea lipalo tsa tlhahlobo, mohlala, ka ts'ebetso ea Riemann zeta, e sebelisa mekhoa e rarahaneng ea tlhahlobo ho ithuta kabo ea linomoro tsa mantlha.
8. Qetello
Tlhahlobo e rarahaneng e fana ka motsoako oa botle le matla: mehopolo e bonahalang e le bonolo joalo ka linomoro tse rarahaneng ha e le hantle e hlahisa khopolo-taba e hlophisitsoeng hantle ho feta tlhahlobo ea 'nete. Tlhokahalo ea holomorphic e beha litšobotsi tse sa tloaelehang mesebetsing, joalo ka ho atolosoa ho ba letoto la matla, ho latela khopolo-taba e matla ea bohlokoa, le ho ba le kamano e tebileng le jiometri ka limmapa tse lumellanang. Ho sa le joalo, lisebelisoa tse kang letoto la Laurent le khopolo-taba ea masala li etsa hore lipalo tse rarahaneng tsa bohlokoa li hlophisehe haholoanyane ebile li be ntle.
Qetellong, tlhahlobo e rarahaneng ha se feela thuto ea "lipalo tse inahaneloang," empa ke puo e feto-fetohang bakeng sa ho utloisisa meaho ea lipalo le liketsahalo tsa tlhaho. Ka lebaka la matla a likhopolo-taba tsa eona le bophara ba ts'ebeliso ea eona, tlhahlobo e rarahaneng e ntse e le e 'ngoe ea libaka tsa bohlokoa le tse thabisang tsa lipalo tsa sejoale-joale.
