De Moivre’s Theorem Roots of a Complex Number Hyperbolic Function

1. GANDHINAGAR INSTITUTE OF
TECHONOLOGY(012)
SUBJECT : Complex Variables & Numerical Methods
(2141905)
Active Learning Assignment on the topic of
“Complex Numbers & Functions”
BE Mechanical Sem:4
Prepared By: Yash D. Pandya
Guided By : Prof. Mansi Vaishnani

2. OVERVIEW
 De Moivre’s Theorem
 Roots of a Complex Number
 Example
 Hyperbolic Function
 Reference

3. De Moivre’s Theorem
 If n is a rational number, then the value or one of the values of
.sincosis)sin(cos  nini n

Corollary 1
Corollary 2
Corollary 3
 sincos)sin(cos ini n
 
 nini n
sincos)sin(cos 
 nini n
sincos)sin(cos  

4. Roots of a Complex Number
 De Moivre’s therom is also useful for finding roots of a complex number.
 If n is any positive integer, then by De Moivre’s therom…


sincossincossincos i
n
ni
n
n
n
i
n
n



















 
    




sin2sinandcos2cos
function;trictrigonometheofnatureperiodicbyobtainedbemayrootsremainingThe
sincossincos
;sincosofrootntheofoneissincosThus,
1



kk
n
i
n
i
i
n
i
n
n
th

5. Roots of a Complex Number
General Form
      





 





 


n
k
i
n
k
kiki nn


2
sin
2
cos
2sin2cossincos
11
   
above.asorderinrootssamethegivek willofluesfurther vaThe
)1(2
sin
)1(2
cos,1
.........
.........
2
sin
2
cos,1
sincos,0
For
follows;As.1...,3,2,1forsincosofrootstheallgiveswhich
1



 



 






 





 



n
n
i
n
n
nk
n
i
n
k
n
i
n
k
nki n





6. Example:1
 Solve and find its all roots.014
z
 
.3,2,1,0;
4
12
sin
4
12
cos
)2sin()2cos(
)sin(cos
)1(
4
1
4
1
4
1











 





 




k
k
i
k
kik
i
z



)1(
2
1
22
1
4
7
sin
4
7
cos,3
)1(
2
1
22
1
4
5
sin
4
5
cos,2
)1(
2
1
22
1
4
3
sin
4
3
cos,1
)1(
2
1
22
1
4
sin
4
cos,0
For
4
3
2
1
i
i
izk
i
i
izk
i
i
izk
i
i
izk









7. Example:2
 Find all roots of .3
8i
iz
iz
iz
80
08
then,8Let
3
3
3



 We have = /2 and r = |z| = 8….therefore, 
.2,1,0;
6
14
sin
6
14
cos2
2
2sin
2
2cos8z
Hence,
2
2sin
2
2cos8
2
sin
2
cos8
)sin(cos
3
1
3
1
3











 





 















































k
k
i
k
kik
kik
i
irz












8. iiiizk
i
i
izk
i
i
izk
2)0(2
2
3
sin
2
3
cos2
6
9
sin
6
9
cos2,2For
3
22
3
2
6
5
sin
6
5
cos2,1For
3
22
3
2
6
sin
6
cos2,0For
3
2
1










































 Above the different 3 value of Z are roots of the given function.

9. Hyperbolic Function
Defination of Hyperbolic Function
 
 
 
 
R,x
ee
ee
(x)
(x)
R,x
ee
x
x
R,x
ee
x
x
xx
xx
xx
xx














tanh
as...definedisandtanhbydenotedisxoftangentHyperbolic
2
cosh
as...definedandcoshbydenotedisxofcosineHyperbolic
2
sinh
as...definedandsinhbydenotedisxofsineHyperbolic

10. Hyperbolic Function
Relation between circular & Hyperbolic Function
)1.(..........)sinh()sin(,
)sinh(
2
2
2
2
)sin(
...
2
sin
)()(
xiixTherefore
xi
ee
i
i
ee
i
ee
i
ee
ix
equationaboveinixbyxreplacing
i
ee
x
xx
xx
xx
ixiixi
ixix







 















11. Hyperbolic Function
Relation between circular & Hyperbolic Function
.cot)coth(
,cothcot
,sec)(sec
secsec
,cos)(cos
coscos
,tan)tanh(
tanhtan
,cos)cosh(
coshcos,
xiix
(x)i(ix)
xixh
h(x),(ix)
ecxiixech
ech(x),iec(ix)
xiix
(x),i(ix)
xix
(x),(ix)Similarly











12. Hyperbolic Function
Hyperbolic Identities
   
)(sinh4)sinh(3)3sinh(
)cosh()sinh(2)2sinh(
1)(coth)(cos
)(tanh1)(sec,
.1)(sinh)(cos
1)cosh()sinh(
1)(cos)(sin
...,
1cossin
3
22
22
22
22
22
22
xxx
xxx
xxech
xxhSimilarly
xx
xxi
ixix
aboveinixbyxreplacingNow
xx









13. Hyperbolic Function
Hyperbolic Function for Complex Number
)cosh(
)sinh(
)tanh(
,isfunctionOther
2
)sinh(
as,definedanddenotedisoffunctionsinehyperboliccomplexThe
2
)cosh(
as,definedanddenotedisoffunctioncosinehyperboliccomplexThe
z
z
z
ee
z
z
ee
z
z
zz
zz








14. Example:3
 Prove that ).1ln(sin 21
ziziz 
 
 
).1ln(sin
)1ln(
)1ln(
1ln
1)(ln
)(sinh
)sinh(
sin
,bysidesbothgMultiplyin
.sinsin
Let,
21
2
22
2
2
1
1
ziziz
ziziw
ziziiw
ziziw
iziziw
iziw
iwiz
wiiz
i
wzwz













15. Thank You