8/12/2019 Comment rsoudre une quation complexe ?
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GakfcOlehrcsKlepncxcs
EtflmcsctJoojncs
fttp0//cxl
s8ejtf.grcc.gr/
[jvlar rslumrc uoc qujtalo klepncxc.
Xjppcn mc klurs
Rrlprat
Mcux olehrcs klepncxcs z ct z slot djux sa, ct scuncecot sa 0
Xc(z) 5Xc(z)Ae(z) 5Ae(z)
lu coklrc|z|5|z|
Jrd(z) 5Jrd(z)
Moatalo
[lat nqujtalo (C) 0 jz8 +hz + k 5 2 jvck j, hct k rcns ct j5
2.Lo jppcnnc 5 h8 =jk nc maskraeaojot mc nqujtalo (C).
Etflmc
Mjos nc kjs l 2, ncs slnutalos mc (C) slot rcnncsct kcst nj
rslnutalo knjssaquc mc kc typc mqujtalos. Mjos nc kjs l > 2, ncs
slnutalos mc (C) slot klepncxcs ct klo`uducs 0
z: 5h+a
8 ct z8 5 z:5h
a
8
Cxcrkakc : (Jeraquc [um 82:;)
Lo klosamrc nqujtalo klepncxc (C) 0 z8 8z; + = 5 2.Xslumrc
nqujtalo (C) mjos ncoscehnc K mcs olehrcs klepncxcs.
Cxcrkakc 8 (Grjokc 8224)
Lo klosamrc nqujtalo (C) 0 z; (= +a)z8 + (:; + =a)z :;a5 2 l z
cst uo olehrc klepncxc.:. Melotrcr quc nc olehrc klepncxc a cst
slnutalo mc kcttc qujtalo.
8. Mtcreaocr ncs olehrcs rcns j, hct ktcns quc, plur tlut olehrc
klepncxc z lo jat 0
z;
(= +a)z8
+ (:; + =a)z :;a5 (z a) jz8 +hz + k .
;. Co mmuarc ncs slnutalos mc nqujtalo (C).
Cxcrkakc :Lo rslut nqujtalo (C) 0 z8 8z; + = 5 2 9(C)5
8;8=:= 5 :8:7 5= > 2. Nquj-talo jmect mlok mcux slnutalos
klepncxcs klo`uducs 0
z: 5 8;+a(=)
8 5
8
; + 8a
8 5
; +a.
ct z8 5 z: 5
; a.
Cxcrkakc 8
:. Rlur rplomrc kcttc qucstalo, an sut mc rcepnjkcrz pjr a mjos
nqujtalo (C) 0
a; (= + a)a8 + (:; + =a)a :;a 5a + = + a = + :;a :;a 5 2 mloka
cst slnutalo mc (C).8. Rlur kcttc qucstalo knjssaquc , lo mlat
mjhlrm mvcnlppcr ncxprcssalo gjktlrasc mc (C), puas amcotacr
ncs
rcns j, h ct k co rslnvjot uo systec mqujtalo 0 (z a)(jz8 +hz +
k) 5 jz; + (h ja)z8 + (k ha)z ak.Mcux plnyoecs slot djux sa ct
scuncecot sa ncs klckacots slot djux. Lo lhtacot nc systec 0
j 5 :h ja 5 = ak ha 5 :; + =aak 5 :;a
j 5 :k 5 :;h a 5 = a:; ha 5 :; + =a
j 5 :h 5 =k 5 :;
mlok z; (= + a)z8 + (:; + =a)z:;a 5 (z a)(z8 =z+ :;).;. Nqujtalo
(C) skrat(z a)(z8 =z+ :;) 5 2.
Mjos K, uo prlmuat mc gjktcurs cst oun sa ct scuncecot sa nuo
mcs gjktcurs cst oun.
Kfcrkflos ncs slnutalos qua joouncot nc mcuxaec ecehrc 0 kcst
marc rslumrc nqujtalo z8 =z+ :; 5 2.Nc maskraeaojot mc kcttc
qujtalo vjut 0 5 (
=)8
=
:
:; 5
;7 > 2.
An y j mlok mcux rjkaocs klepncxcs klo`uducs z: 5(=) + a(;7)
8 5= + 7a
8 5 8+ ;act z85 z: 5 8;a.
Ncoscehnc mcs slnutalos cst 0[5{a 9 8 + ;a 9 8 ;a}.
