complex numbers 5 6 7 handout
TRANSCRIPT
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8/13/2019 Complex Numbers 5 6 7 Handout
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Complex exponentialTrigonometric and hyperbolic functions
Complex logarithmComplex power function
Chapter 13: Complex NumbersSections 13.5, 13.6 & 13.7
Chapter 13: Complex Numbers
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8/13/2019 Complex Numbers 5 6 7 Handout
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Complex exponentialTrigonometric and hyperbolic functions
Complex logarithmComplex power function
DenitionProperties
1. Complex exponential
The exponential of a complex number z = x + iy is dened as
exp(z ) = exp( x + iy ) = exp( x ) exp( iy )= exp( x ) (cos( y ) + i sin(y )) .
As for real numbers, the exponential function is equal to itsderivative, i.e.
d
dz exp(z ) = exp( z ). (1)
The exponential is therefore entire.
You may also use the notation exp(z
) = e z
.
Chapter 13: Complex Numbers
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Complex exponentialTrigonometric and hyperbolic functions
Complex logarithmComplex power function
DenitionProperties
Properties of the exponential functionThe exponential function is periodic with period 2 i : indeed,for any integer k Z ,
exp(z + 2 k i ) = exp( x ) (cos( y + 2 k ) + i sin(y + 2 k ))= exp( x ) (cos( y ) + i sin(y )) = exp( z ).
Moreover,
| exp(z )| = | exp(x )| | exp(iy )| = exp( x ) cos2(y ) + sin 2(y )= exp( x ) = exp ( e (z )) .
As with real numbers,exp(z 1 + z 2) = exp( z 1) exp(z 2);
exp(z ) = 0.
Chapter 13: Complex Numbers
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Complex exponentialTrigonometric and hyperbolic functions
Complex logarithmComplex power function
Trigonometric functionsHyperbolic functions
2. Trigonometric functions
The complex sine and cosine functions are dened in a waysimilar to their real counterparts,
cos(z ) = e iz + e iz
2 , sin(z ) =
e iz e iz
2i . (2)
The tangent, cotangent, secant and cosecant are dened asusual. For instance,
tan( z ) = sin(z )cos(z )
, sec(z ) = 1cos(z )
, etc.
Chapter 13: Complex Numbers
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Complex exponentialTrigonometric and hyperbolic functions
Complex logarithmComplex power function
Trigonometric functionsHyperbolic functions
Trigonometric functions (continued)
The rules of differentiation that you are familiar with stillwork.
Example:Use the denitions of cos(z ) and sin(z ),
cos(z ) = e iz + e iz
2 , sin(z ) =
e iz e iz
2i .
to nd (cos(z )) and (sin(z )) .
Show that Eulers formula also works if is complex.
Chapter 13: Complex Numbers
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Complex exponentialTrigonometric and hyperbolic functions
Complex logarithmComplex power function
Trigonometric functionsHyperbolic functions
3. Hyperbolic functionsThe complex hyperbolic sine and cosine are dened in a waysimilar to their real counterparts,
cosh(z ) = e z + e z
2 , sinh(z ) =
e z e z
2 . (3)
The hyperbolic sine and cosine, as well as the sine and cosine,are entire.
We have the following relations
cosh(iz ) = cos( z ), sinh(iz ) = i sin(z ),(4)
cos(iz ) = cosh( z ), sin(iz ) = i sinh(z ).
Chapter 13: Complex Numbers
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8/13/2019 Complex Numbers 5 6 7 Handout
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Complex exponentialTrigonometric and hyperbolic functions
Complex logarithmComplex power function
DenitionPrincipal value of ln( | z | )
4. Complex logarithmThe logarithm w of z = 0 is dened as
e w = z .
Since the exponential is 2 i -periodic, the complex logarithm ismulti-valued.
Solving the above equation for w = w r + iw i and z = re i
gives
e w = e w r e iw i = re i = e w r = r w i = + 2p ,
which implies w r = ln( r ) and w i = + 2p , p Z .
Therefore,ln(z ) = ln( |z |) + i arg(z ).
Chapter 13: Complex Numbers
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Complex exponentialTrigonometric and hyperbolic functions
Complex logarithmComplex power function
DenitionPrincipal value of ln( | z | )
Principal value of ln(z
)
We dene the principal value of ln(z ), Ln(z ), as the value of
ln(z ) obtained with the principal value of arg(z ), i.e.
Ln(z ) = ln( |z |) + i Arg(z ).
Note that Ln( z ) jumps by 2 i whenone crosses thenegative real axisfrom above.
y
x 0
Arg( z )=
1
1
Arg( z )> -
The negative real axis is called a branch cut of Ln(z ).
Chapter 13: Complex Numbers
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Complex exponentialTrigonometric and hyperbolic functions
Complex logarithmComplex power function
DenitionPrincipal value of ln( | z | )
Principal value of ln(z
) (continued)
Recall thatLn(z ) = ln( |z |) + i Arg(z ).
Since Arg(z ) = arg( z ) + 2 p , p Z , we therefore see thatln(z ) is related to Ln(z ) by
ln(z ) = Ln( z ) + i 2p , p Z .
Examples:
Ln(2) = ln(2), but ln(2) = ln(2) + i 2p , p Z .Find Ln( 4) and ln( 4).
Find ln(10 i ).
Chapter 13: Complex Numbers
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8/13/2019 Complex Numbers 5 6 7 Handout
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Complex exponentialTrigonometric and hyperbolic functions
Complex logarithmComplex power function
DenitionPrincipal value of ln( | z | )
Properties of the logarithmYou have to be careful when you use identities like
ln(z 1z 2) = ln( z 1)+ln( z 2), or ln z 1z 2= ln( z 1) ln(z 2).
They are only true up to multiples of 2 i .
For instance, if z 1 = i = exp( i / 2) and z 2 = 1 = exp( i ),
ln(z 1) = i 2
+ 2 p 1i , ln(z 2) = i + 2 p 2i , p 1, p 2 Z ,
andln(z 1 z 2) = i
32
+ 2 p 3i , p 3 Z ,
but p 3 is not necessarily equal to p 1 + p 2.Chapter 13: Complex Numbers
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Complex exponentialTrigonometric and hyperbolic functions
Complex logarithmComplex power function
DenitionPrincipal value of ln( | z | )
Properties of the logarithm (continued)Moreover, with z 1 = i = exp( i / 2) and z 2 = 1 = exp( i ),
Ln(z 1) = i 2 , Ln(z 2) = i ,
andLn(z 1 z 2) = i
2
= Ln( z 1) + Ln( z 2).
However, every branch of the logarithm (i.e. each expressionof ln(z ) with a given value of p Z ) is analytic except at thebranch point z = 0 and on the branch cut of ln( z ). In thedomain of analyticity of ln(z ),
d dz
(ln(z )) = 1z . (5)
Chapter 13: Complex Numbers
C l i l
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8/13/2019 Complex Numbers 5 6 7 Handout
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Complex exponentialTrigonometric and hyperbolic functions
Complex logarithmComplex power function
Denition
5. Complex power function
If z = 0 and c are complex numbers, we dene
z c = exp ( c ln(z ))= exp ( c Ln(z ) + 2 pc i ) , p Z .
For c C , this is again a multi-valued function, and we denethe principal value of z c as
z c = exp ( c Ln(z ))
Chapter 13: Complex Numbers