COMPLEX NUMBERS AND ELEMENTARY FUNCTIONS OF COMPLEX VARIABLES
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1 COMPLEX NUMBERS AND ELEMENTARY FUNCTIONS OF COMPLEX VARIABLES DEFINITION OF A COMPLEX NUMBER: A umbr of th form, whr = (, ad & ar ral umbrs s calld a compl umbr Th ral umbr, s calld ral part of whl s calld magar part of A ral umbr u ca bprssd b u = u + 0 Hcvr ral umbr s a compl umbr but th covrs s ot tru Proprts : ( Th cojugat of a compl umbr s dotd & dfd b = ( If th R ( = = + ( If ad th, ad Im ( = = - (I Sum: + = ( + ( = ( ( (II Dffrc: - = ( - ( = ( ( (III Product: = ( ( = ( (IV Quott : ( = ( ( s calld modulus or absolut valu of ( Prpard b MrZalak Patl Lcturr, Mathmatcs
2 GEOMETRICAL REPRESENTATION OF COMPLEX NUMBER: Lt b a compl umbr w ca us ordr par of ral umbrs (, to dot compl umbr th pla Thus th ordr par of ral umbrs (, ca b dtfd wth a pot th pla; wth such a dtfcato, th pla s calld th compl pla Th horotal as s calld th ral as Th vrtcal as s calld th magar as = (, = + 0 r = Z - - FIG 0 FIG 0 POLAR FORM OF A COMPLEX NUMBERS: Lt b a compl umbr, hr & ar Cartsa coordats of I prvous dscusso If w dclar (0,0=0+0 as pol ad X as as polar as th usg gomtr w wll hav = r cos ad = r s Hc, = r cos + r s = r (cos + s Also r ca b cosdr as dstac of pot from pol ad ca b otad usg ta - (slop of th l passg through pol ad (, r ad = ta rlatd to polar form r s calld th absolut valu or modulus of ad s calld argumt or ampltud of ad t s dotd b arg Hc, arg = = ta s cos r r Th valu of arg whch ls th trval s calld Prcpal valu of th argumt of ( 0 Gral valu of th arg ca b prssd as + ( Prpard b MrZalak Patl Lcturr, Mathmatcs
3 DE MOIVRE S THEOREM (ol statmt: STATEMENT: Lt b a ratoal umbr th th valu or o of th valu s of (cos + s s cos + s Rsults : W wll dot cos + s b c s ( If = cos + s - (cos + s cos( cos s s ( th - Thus - = cs ( = = cos - s cs ( ( cs cs ( ( cs = = cos - s (cs ( ( cs ( cs cs ( cs ( cs( cs ROOTS OF COMPLEX NUMBER: D Movr s thorm ca b usd to fd all -roots (valus of a compl umbr Sc, s = s (k + ad cos = cos (k +, Or cs = cs (k +,Whr k s a tgr ( cs [ cs (k ] cs k, whr k 0,,,,, Thus, w gt roots of [ cs (k ] ( Prpard b MrZalak Patl Lcturr, Mathmatcs
4 APPLICATION OF DE MOIVRE S THEOREM TO TRIGONOMETRICAL IDENTITTIES: Usg D Morv s thorm ad corrspodg rsults w hav followg two rsults ( I Epaso of s, cos powrs of s, cos,wh s a postv tgr B D Morv s thorm cos + s = (cos + s Epadg RHS b Bomal thorm ad quatg ral &magar parts w gt rqurd pasos ( II Epaso of s, cos ad s m cos m powrs of s, cos, wh s a postv tgr Lt cos + s th cos + s cos + s ad cos - s hc, cos ad s cos ad s usg abov rsults w ca pad powrs of s or cos or thr products a srs of coss or ss of multpls of EULER S FORMULAE : Sc for a valu of, w kow that!!! s!! cos!! Usg abov srs, w gt ( ( (!!! cos!! (!! s (! Smlarl cos s ( Formula gv b ( & ( ar calld Eulr s formula ( Prpard b MrZalak Patl Lcturr, Mathmatcs
5 Abov all dscusso lad us to followg rsult EXPONENTIAL FORM OF A COMPLEX NUMBER : From prvous rsult cos s For a compl umbr, r cos s cartsa form polar form r Epotal form Thus th form = r s calld Epotal form of a compl umbr CIRCULAR FUNCTIONS: From Eulr s formula, If s a ral or compl cos W gt, Hc, s ta = cot = cosc sc = cos ad cos HYPERBOLIC FUNCTIONS: s s Dfto: If s a ral or compl Hprbolc s ad cos of s dotd ad dfd b sh ad cosh Othr Hprbolc fuctos ar dfd as tah =, coth =, cosch, sch = ( Prpard b MrZalak Patl Lcturr, Mathmatcs
6 PROPERTIES: ( Sh & cosh ar odd & v fucto rspctvl sh(- = sh ad cosh(- = cosh ( Idtt: cosh sh sc h tah coht cosch ( tah = sh cosh cosh, coth =, sch =, cosch = sh cosh sh ( sh A sh Acosh A ( cosh A cosh A sh A = cosh A = + sh A tah A (6 taha = tah A (7 sh A sh A sh A, cosh A cosh A cosh A (8 cosh A cosh A cosh A (9 cosh A cosh A sh A (0 cosh A (cosh A, sh A (cosh A ( sh( A B sh Acosh B cosh A sh B ( sh( A B sh Acosh B cosh A sh B ( cosh( A B cosh Acosh B sh A sh B ( cosh( A B cosh Acosh B sh A sh B tah A tah B tah A tah B ( tah (A+B =, tah (A - B = tah Atah B tah Atah B RELATION BETWEEN CIRCULAR & HYPERBOLIC FUNCTIONS: s = sh, cos = cosh, ta = tah sh( = s, cosh( = cos, tah( = ta sch( = sc, cosch( = - cosc, coth( = - cot (6 Prpard b MrZalak Patl Lcturr, Mathmatcs
7 INVERSE HYPERBOLIC FUNCTIONS: If sh = th s calld vrs hprbolc s of ad s dotd b = sh - Smlarl w ca df cosh -, tah -, cosch -, sch -, coth - For ral valu of w ca prov th followg rsults ( s h ( ( cos h ( ( ta h REAL AND IMAGINARY PARTS OF CIRCULAR FUNCTIONS: ( s ( s cos cos s s cosh cos sh ( cos( cos cos s s cos cosh s sh ( ta( s ( cos( s ( cos( cos ( cos( s s( cos cos( s sh cos cosh s cos cosh sh cos cosh REAL AND IMAGINARY PARTS OF HYPERBPLIC FUNCTIONS: ( s h( sh cos cosh s ( cos( cosh cos sh s ( ta( sh cosh cos s cosh cos (7 Prpard b MrZalak Patl Lcturr, Mathmatcs
8 LOGARITHM OF A COMPLEX NUMBER: Lt b a compl umbr ad lt = r cos ad = r s, th r ad = ta Hc, = r cos + r s = r (cos + s = r r ( ( ta ( Th valu of arthm gv b ( & ( s calld th prcpal valu of th arthm of Also, ( = r cos + r s = r (cos + s = r [cos ( + + s ( + ] ( = r Log r ( ( ta ( Th valu of arthm gv b ( & ( s calld th gral valu of th arthm of Thus th gral valu of th arthm s a mult-valud fucto whl th Prcpal valu of th arthm s a sgl-valud fucto If w put = 0 ( w wll gt th prcpal valu of th arthm of RELATION BETWEEN PRINCIPAL AND GENERAL VALUE OF LOGARITHM : From ( & ( w ca wrt Log ( COMPLEX EXPONENT : To fd Z C w wll us Z C = C Z (8 Prpard b MrZalak Patl Lcturr, Mathmatcs
9 Solv Followg Problms: (Class room work Prov that (s + cos cos s cos s cos s Prov that cos s 7 8 cos s cos s Prov that : (I ( ( cos (II ( ( cos 6 (III ( ( Prov that : cos s (s s cos s (s s 8 s s cos Prov that : 8 8 s cos 8 8 If cos, cos ad p q r Prov that ( cos p p q r m ( cos m m If a = cs ad b = cs Prov that : ( cos ( - a b b a cos q cos a b ( s( - b a Show that th modulus ad prcpal valu of th argumt of Prov that ar ad 6 rspctvl Fd all th roots of th followg: ( 6 ( - ( - 6 ( ( r f f k k (9 Prpard b MrZalak Patl Lcturr, Mathmatcs
10 Fd all th valus of ( ad show that thr cotud product s + Solv thquato 0 ad fd whch of ts roots satsf thquato 0 Prov that th th roots of ut ar gomtrc progrsso hc prov that th sum of ths roots s ro Us D Movr s thorm to solv followg: 8 8 ( 0 ( 0 ( 7 ( 0 (v 0 6 Prov that : cos6 cos 8cos 8cos s 7 6 Prov that : 7 6 s s 6 s s ta 0 ta ta 7 Prov that : ta 0 ta ta 8 If = cos, prov that ( + cos8 = ( 6 9 Prov that : cos cos 8 cos 0 Prov that : s 7 s 6 s 7s s 7 Prov that : 7 cos s s s0 s 8 0s 6 Prov that cosh sh cosh sh If s = ta h, prov that ta = s h Show that cosch + coth = cot h If = ta prov that : ( ta h ( cos h cos ta 6 Prov that : sh (ta ta 7 Prov that : cos h cosh s 0s (0 Prpard b MrZalak Patl Lcturr, Mathmatcs
11 8 Prov that: ta a a a 9 Prov that : ( ta h ( s h lm s h cos h ta 0 Prov that : s h a If s, ( cosc sc Prov that: ( sc h cosch If s (cos s ( cosh cos Prov that: ( ta tah cot Prov that: cos Sparat ta a b to ral ad magar parts Prov that s cosc cot 6 Fd gral ad prcpal valu of th ( - ad ( + 7 Prov that: ( ( + ta = sc + ( (+ ( cos ( ( cos s( (v ta cot tah s( cosh cos (v cos( ta (v ta ab (v ta a b ta tah ( Prpard b MrZalak Patl Lcturr, Mathmatcs
12 8 Sparat followg to ral ad magar parts, ad fd modulus ad argumt: ( ( a ( v ( v ( 9 Prov that s wholl ral ad show that th valu of form a gomtrc progrsso 0 Fd all roots of thquato s h = ( Prpard b MrZalak Patl Lcturr, Mathmatcs
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