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![An important interpretation regarding multiplication
given by polar form of complex number
z1 = r1 (cos θ1 + i sin θ1 )
z2 = r2 (cos θ2 + i sin θ2 )
z1z2= r1 r2 (cos θ1 + i sin θ1 ) (cos θ2 + i sin θ2)
=r1r2(cos θ1cos θ2 - sin θ1sin θ2)+i(sin θ1cos θ2+cos θ1sin θ2)
= r1r2 [cos(θ1 + θ2)+i sin (θ1 + θ2)] = r1r2 cis (θ1 + θ2)](https://image.slidesharecdn.com/1complexnumbers-150115045600-conversion-gate01/75/1-complex-numbers-8-2048.jpg)
![Roots of a complex number
n
θ
sini
n
θ
cos)θsiniθ(cos n ++++====++++
1
If n is a positive integer than is one of the root of
that is
n
θ
sini
n
θ
cos ++++
n)θsiniθ(cos
1
++++
nn
++++
++++
++++
====
++++++++++++====++++
n
θπk
sini
n
θπk
[cos
)]θπksin(i)θπk[cos()θsiniθ(cos
nn
22
22
11
Remaining roots can be obtained by periodic nature of sine and cosine
It gives all roots of for K = 0, 1, 2, 3, …(n – 1)n)θsiniθ(cos
1
++++](https://image.slidesharecdn.com/1complexnumbers-150115045600-conversion-gate01/75/1-complex-numbers-14-2048.jpg)
![LOGARITHMIC FORM OF COMPLEX NUMBER
zLogwze,Cw,zIf e
w
====⇒⇒⇒⇒====∈∈∈∈
w)z(Log
Ik,kiπw)z(Log
ze,Now
e
iπkw
====
∈∈∈∈++++====
====++++
2
2
iπk)iyxlog()iyx(Log
reiyxzAs θi
2++++++++====++++
====++++====
iπk)iyxlog()iyx(Log 2++++++++====++++
iπktani)yxlog(
iπkθiyxlog
iπk)elog()rlog(
iπk)relog(
θi
θi
2
2
1
2
2
2
122
22
++++++++++++====
++++++++++++====
++++++++====
++++====
−−−−
x
y
x
y
m 122
2
2
1 −−−−
++++====++++++++====++++ tanπk)]iyx(Log[I),yxlog()]iyx(LogRe[](https://image.slidesharecdn.com/1complexnumbers-150115045600-conversion-gate01/75/1-complex-numbers-20-2048.jpg)
Uploaded bygandhinagar
1 complex numbers
This document discusses complex numbers including: 1. Defining complex numbers and their algebraic properties such as addition, subtraction, multiplication and division. 2. Geometrically representing complex numbers in Cartesian and polar forms. 3. Key concepts such as the absolute value, distance between complex numbers, and the interpretation of multiplication in polar form. 4. De Moivre's theorem and its expansion along with examples of evaluating complex numbers and finding roots of complex numbers using this theorem. 5. Exponential and logarithmic forms of representing complex numbers.
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1 complex numbers
- 1.
- 2. In this unitwe will discuss …… Introduction and basic definition of Complex numbers. Algebraic properties of Complex numbers. De Moivre’s theorem and its expansion. Exponential form of Complex numbers. Logarithm of a Complex numbers. Hyperbolic and Inverse hyperbolic functions.
- 3. DEFINITION OF COMPLEXNUMBERS i=−1 Complex number Z = a + bi is defined as an ordered pair (a, b), where a & b are real numbers and . a = Re (z) b = im(z)) Two complex numbers are equal iff their real as well as imaginary parts are equal Complex conjugate to z = a + ib is z = a - ib (0, 1) is called imaginary unit i = (0, 1).
- 4. ALGEBRA OF COMPLEXNUMBERS Addition and subtraction of complex numbers is defined as idbcadicbia )()()()( ±+±=+±+ Multiplication of complex numbers is defined as iadbcbdacdicbia )()())(( ++−=++ Division of complex numbers is defined asDivision of complex numbers is defined as i dc adbc dc bdac dic bia 2222 )( )( + −+ + += + + Relation between z and z (((( )))) )Z( Z Z Z Z ;ZZZZ ,zzz;zz;z)z( i zz zIm, zz zRe 0 22 2 2 1 2 1 2121 2 ≠≠≠≠==== ==== ============ −−−− ==== ++++ ====
- 5. GEOMETRICAL REPRESENTATION OFCOMPLEX NUMBERS If z = a + ib, is a complex number than in cartesian form it is as good as (a, b) For polar form, let us take a = r cos θ and b = r sin θ z = rcos θ + i rsin θ = r(cos θ + i sin θ), = r cis θ πθπ b tanθ π bar ≤≤≤≤<<<<======== ±±±±±±±±====++++==== ====++++==== −−−− -, a Arg(z) ...2.........1,0,Kk,2Arg(z)arg(z) ,z 1 22
- 6. Geometrically, IzI isdistance of point z from origin. θ is directed angle from positive X – axis to (0, 0) – (a, b) θ between - π < θ < π is called principal argument and denoted by Arg (z)
- 7. The absolute valueor modulus o the number z = a + bi is denoted by |z| given by 22 baz += 2121 )inequalitytriangular(zzzz ++++≤≤≤≤++++ ABSOLUTE VALUE & DISTANCE Distance between the points z1 = a1+b1i and z2 = a2+b2i is denoted by 2 21 2 2121 )()( bbaazz −+−=− 1212 zzzz −−−−≤≤≤≤−−−−
- 8. An important interpretationregarding multiplication given by polar form of complex number z1 = r1 (cos θ1 + i sin θ1 ) z2 = r2 (cos θ2 + i sin θ2 ) z1z2= r1 r2 (cos θ1 + i sin θ1 ) (cos θ2 + i sin θ2) =r1r2(cos θ1cos θ2 - sin θ1sin θ2)+i(sin θ1cos θ2+cos θ1sin θ2) = r1r2 [cos(θ1 + θ2)+i sin (θ1 + θ2)] = r1r2 cis (θ1 + θ2)
- 9. The modulus ofthe product is product of the moduli The argument of the product is sum of the argument |z1z2|=|z1 || z2| arg (z1z2)= arg z1 + argz2 z1 z2 θ1 + θ2 θ2 θ1 z1 z2
- 10. EXAMPLES Q. Find thecomplex conjugate of i i −−−− ++++ 1 23 Q. Determine Region in z – plane represented by ) z z (argand)zz(arg ,izandizIf 2 1 21 21 32231 ++++====++++−−−−====Q. 1<|z-2|<3
- 11. Q. Express the complexnumber in polar form and find the principle argument. i++++−−−− 3 Q. Express the complex number in polar form and find the principle argument. 31 i++++
- 12. De Moivre’s Theorem Ifn is a rational number than the value or one of the values of (cos θ + i sin θ)n is cos nθ + i sin nθ. In particular, (cos θ + i sin θ)n = cos nθ + i sin nθ for n = 0, ±1, ±2 …………. For any complex number z = r e i θ and n = 0, ±1, ±2 …………., we have zn = rn e i nθ
- 13. Q. 9090 3131 )i()i(Evaluate−−−−++++++++ θsiniθcos )θsiniθ(cos)θsiniθ(cos )θsiniθ(cos)θsiniθ(cos thatovePr 77 5533 22 3 12 23 2 ++++==== −−−−−−−− −−−−++++ Q. Examples - De Moivre’s Theorem )θsiniθ(cos)θsiniθ(cos 5533 −−−−−−−− Q. 4311 311 58 46 i )i()i( )i()i( thatovePr ==== ++++−−−− −−−−++++ Q. −−−− −−−−====−−−−++++++++++++++++ ++++ 2424 211 1 θnπn cos θπ )θcosiθsin()θcosiθsin( nnn Cos n
- 14. Roots of acomplex number n θ sini n θ cos)θsiniθ(cos n ++++====++++ 1 If n is a positive integer than is one of the root of that is n θ sini n θ cos ++++ n)θsiniθ(cos 1 ++++ nn ++++ ++++ ++++ ==== ++++++++++++====++++ n θπk sini n θπk [cos )]θπksin(i)θπk[cos()θsiniθ(cos nn 22 22 11 Remaining roots can be obtained by periodic nature of sine and cosine It gives all roots of for K = 0, 1, 2, 3, …(n – 1)n)θsiniθ(cos 1 ++++
- 15. Examples: Q. Solve Z4+ 1 = 0 )i(),i(),i(),i( −−−−−−−−−−−−++++−−−−++++ 1 2 1 1 2 1 1 2 1 1 2 1 Q. Find fifth root of i++++−−−− 3Q. Find fifth root of i++++−−−− 3 ++++ ++++ ++++ ++++ ++++ 30 53 30 53 2 30 41 30 41 2 30 29 30 29 2 30 17 30 17 2 66 2 51 5151 5151 π sini π cos , π sini π cos, π sini π cos , π sini π cos, π sini π cos
- 16. Q. Solve theequation x 4 – x3 + x2 – x +1 = 0 using De Moivre’s theorem. (((( )))) ++++++++ ++++ ++++ 77 22 5 3 5 3 2 55 2 5151 5151 , π sini π cos,πsiniπcos , π sini π cos, π sini π cos (((( )))) ++++ ++++++++ 5 9 5 9 2 5 7 5 7 22 51 5151 π sini π cos , π sini π cos,πsiniπcos
- 17. θsini z z,θcos z z 2 1 2 1 ====−−−−====++++ Expansion ofDe Moivre’s Theorem θsin)i( z z,θcos z z nnnn n 2 1 2 1 ==== −−−−==== ++++ θnsini z z,θncos z z n n n n 2 1 2 1 ==== −−−−==== ++++ zz
- 18. Examples: Q. Express Cos6θ in terms of cosines of multiples of θ.
- 19. Let z isa complex number, then ez is called exponential function ez = e x + iy = e x e iy For each y ∈ R , complex number e iy is defined as Known as Euler’s formulayiyeiy sincos += EXPONENTIAL FORM OF COMPLEX NUMBER Known as Euler’s formulayiyeiy sincos += )sin(cos, yiyeeeeeiyxzFor xiyxiyxz +===+= + (((( )))) (((( )))) ysineeIm,ycoseeRe xzxz ======== )zRe(ee),z(imy)earg( xzz ================
- 20. LOGARITHMIC FORM OFCOMPLEX NUMBER zLogwze,Cw,zIf e w ====⇒⇒⇒⇒====∈∈∈∈ w)z(Log Ik,kiπw)z(Log ze,Now e iπkw ==== ∈∈∈∈++++==== ====++++ 2 2 iπk)iyxlog()iyx(Log reiyxzAs θi 2++++++++====++++ ====++++==== iπk)iyxlog()iyx(Log 2++++++++====++++ iπktani)yxlog( iπkθiyxlog iπk)elog()rlog( iπk)relog( θi θi 2 2 1 2 2 2 122 22 ++++++++++++==== ++++++++++++==== ++++++++==== ++++==== −−−− x y x y m 122 2 2 1 −−−− ++++====++++++++====++++ tanπk)]iyx(Log[I),yxlog()]iyx(LogRe[
- 21. Examples: Q. Prove that22 2 ba ab iba iba logitan −−−− ==== ++++ −−−− Q. Find general value of log (-3) and log (- i). Q. Separate real and imaginary parts of 1) log (1+i) 2) log (4+3i)
- 22. Circular functions ofcomplex number i ee xsin, ee xcos ixixixix 22 −−−−−−−− −−−− ==== ++++ ==== Hyperbolic functionsHyperbolic functions xx xxxxxx ee ee xtanh, ee xsinh, ee xcosh −−−− −−−−−−−−−−−− ++++ −−−− ==== −−−− ==== ++++ ==== 22
- 23. HYPERBOLIC AND CIRCULARFUNCTIONS sin h (ix) = i sin x cos h (ix) = cos x tan h (ix) = i tan x cosec h (ix) = -i cosec x sec h (ix) = sec x cot h (ix) = -i cot x
- 24.
- 25. INVERSE HYPERBOLIC FUNCTIONS ++++ ==== −−−−++++==== ++++++++==== −−−− −−−− −−−− x1 an os ln)x(ht )xx(ln)x(hc )xx(ln)x(hsin 1 1 1 1 21 21 ====−−−− x-1 an ln)x(ht 2 1