Download Exponential Form of a Complex Number Lab

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The Exponential Form of a Complex Number Lab
efore we can investigate the exponential form of a complex number, we'll have to do some
preliminary work. Much of it is based on work done by the remarkable mathematician
Leonard Euler.
We are first going to define the number e (through the use of the binomial theorem). Then,
with the help of Euler, we'll develop both the exponential and trigonometric series. This will lead us
directly into Euler's formula, which will allow us to write complex numbers in exponential form.
B
The definition of the number e.
In 1748, Leonhard Euler, a Swiss mathematician, published a work in which he developed a series
whose limiting value was a particular irrational number. In his honor, this irrational number was
called e, the Euler number. The number is the sum of the following infinite series:
1 1 1 1 1
1
e = 1+ + + + + +L + +L
1! 2! 3! 4! 5!
n!
nx
1
This series can be derived using the binomial theorem to expand the expression  1 +  .
n

 n
n
n
n
( x + y ) n =   x n +   x n −1 y +   x n −1 y 2 + L +   y n (Binomial Theorem)
0
1 
2
n
Expanding, we have:
NOTE:
 nx   nx   1   nx   1   nx   1 
 1
1 +  =  0 1 + 1    +  2    +  3    + L
 n
    n    n    n 
2
nx
3
 1  nx(nx − 1)  1  nx(nx − 1)(nx − 2)  1 
= 1 + (nx )   +
  +
  +L
2!
3!
 n
n
 n
2
=
1
1 
2


x  x −  x  x −  x − 
n
n 
n
= 1+ x + 
+ 
+L
2!
3!
nx
 1
lim 1 +  =
n →∞
 n
= 1+ x +
n
 1
lim 1 +  =
n →∞
 n
Revised August 3, 2000
Steve Boast
x2 x3 x4 x5
+ + + + L If we let x = 1,
2! 3! 4! 5!
3
The Exponential Form of a Complex Number Lab
n
1 1 1 1
 1
And so, lim  1 +  = 1 + + + + + L and e can be defined as follows:
n →∞
1! 2! 3! 4!
 n
n
1 1 1 1
 1
e = lim  1 +  = 1 + + + + + L
n →∞
1! 2! 3! 4!
 n
1.
What, if any, restrictions are there on x?
2.
Use the sum and seq features of your graphics calculator to estimate the value of e.
a.
Write the expression that you entered on the calculator and the estimated value for
e.
b.
Compare your answer to the calculator's stored value of e. Why might there be a
difference in the two values for e?
What is the "exponential series?"
nx
1
The exponential series is an immediate consequence of the definition of e and e x = lim  1 +  .
n→ ∞
n

x
n
nx

1 
1
x2 x3 x4
Since e =  lim  1 +   = lim  1 +  = 1 + x + + + + L , the exponential series is
 n→ ∞  n  
n→ ∞
2! 3! 4!
 n


defined as follows:
x
ex = 1+ x +
x2 x3 x4
+
+
+L
2! 3! 4!
Use the sum and seq features of your graphics calculators to estimate the problems below. Write the
expression you used to find the sum of the series and then check your answers by finding the power
of e directly:
1.
e4
Revised August 3, 2000
Steve Boast
2.
e
2
Page 2
The Exponential Form of a Complex Number Lab
What is a "trigonometric series?"
The last preliminary idea we need to explore is that of a trigonometric series. Of course, Euler is
closely associated with this idea, also. The trigonometric series for cos x and sin x are given below:
x2
+
2!
x3
+
sin x = x −
3!
cos x = 1 −
x4
−
4!
x5
−
5!
x6
+
6!
x7
+
7!
x8
−L
8!
NOTE: x must be in radians.
x9
−L
9!
Use the appropriate trigonmetric series to estimate the problems below. Write the expression you
used to find the sum of the series and then check your answers by finding the trigonometric value
directly.
π 
1.
2.
cos ( 300° )
sin  
4
What is the relationship between the exponential and trigonometric series?
If we replace x with iα in the exponential series, we would have the following:
iα
e = 1 + iα +
( iα )
2!
2
+
( iα )
3!
3
+
( iα )
4!
=
4
+L
(α must be in radians)
Recognize anything?
= 1 + iα −
α2 α3 α 4
−i
+
+L
2!
3! 4!
Regroup the terms appropriately.
=
 α2 α4 α6
 

α3 α5 α7
e iα =  1 −
+
−
+ L + i α −
+
−
+ L
and so...
2! 4! 6!
3! 5! 7!

 

This remarkable equation is called Euler's Formula.
e iα = cos α + isin(α )
( )
π
i
1.
Find e 2 using the relationship discovered above. Verify with your graphics calculator.
2.
Find e
5π
i
3
using the relationship discovered above. Verify with your graphics calculator.
Revised August 3, 2000
Steve Boast
Page 3
The Exponential Form of a Complex Number Lab
What is the "exponential" form of a complex number?
We've finally arrived! It's always the preparation that takes most of the time. So, how do we find
the exponential form of a complex number?
1.
2.
Let z = a + bi = [r, "].
a.
What are the restrictions on "?
b.
Find z in trigonometric form.
c.
Find z in exponential form!
Which of the following could represent the equation for the exponential form of the complex
number x + yi? __________
a. x + yi = re
−α 2
b. x + yi = re iα
c. x + yi =
3.
Write 3 + i in exponential form.
4.
Let α = π in the expression e iα = cos (α ) + i sin(α ) .
y

i  tan −1 
x
x2 + y2 e 
( )
d. x + yi = r ln ee
iα
a.
Write the simplified equation in the space below. Check the results with your
graphics calculator.
b.
The resulting equation has been called the most beautiful relationship in mathematics.
How might you explain this rather bold statement?
Revised August 3, 2000
Steve Boast
Page 4
The Exponential Form of a Complex Number Lab
What is the "exponential" form of a complex number? (cont.)
c.
5.
6.
7.
If you set one side equal to zero, you have an equation that expresses a relationship
between five of the most important numbers in mathematics. What are these five
numbers?
Consider the equation e iπ = − 1 .
a.
Take the natural logarithm of both sides. What does this result mean?
b.
Find ln(-6). Verify with your graphics calculator.
Find
(
3 − 3i
)
11
using all of the following methods:
(
3 − 3i
a.
Convert 3 − 3i to polar form and then find
b.
Convert 3 − 3i to exponential form and then find
c.
Convert 3 − 3i to trigonometric form and then find
d.
Use the features of your graphics calculator to find
(
(
)
11
.
3 − 3i
(
)
11
3 − 3i
3 − 3i
)
11
.
)
11
.
directly.
Find the exact value of i i . Would you expect the answer to be a complex number? Hint:
Convert i to exponential form.
Revised August 3, 2000
Steve Boast
Page 5
The Definition of e


1

Expanding 1+ 

n 



1+ 1 

n 

nx
nx
, we have:
 nx 
 nx   1   nx   1 


=  1+     +    
0  1   n   2   n 

2
  nx(nx −1)  
1
= 1+ (nx)  n1  +
2!  n 
 
3
 nx   1 
+    
3  n 

2
+L
 
nx
nx
−
nx
−
(
1)(
2)
1
+
n
3!
 
3
+L
=


1+ 1 
lim
n→∞ 
n 

nx


1

lim 1+ n 
n→∞ 








x  x − 1n  x  x − 1n  x − 2

n
+ 

 +L
= 1+ x + 
2!
3!
=
n


n→∞ 

2
3
4
5
= 1+ x + x + x + x + x +L If we let x = 1,
2! 3! 4! 5!
=
n

And so, lim 1+ 1  = 1+ 1 + 1 + 1 + 1 +L
n 
1! 2! 3! 4!
Definition of e :


n→∞ 






n
e = lim 1+ 1n =1+ 1 + 1 + 1 + 1 +L
1! 2! 3! 4!
Revised August 3, 2000
Steve Boast
Page 6
The Development of Euler's Formula
2
3
4
Letting x = iα in e x = 1+ x + x + x + x +L,
2! 3! 4!
eiα =1+ iα +
 iα 




2!
2
3
+
 iα 




3!
+
 iα 




4!
we would have
4
+L
simplifying,
=
Recognize anything?
2
3 α4
α
α
= 1+ iα − − i + +L
2! 3! 4!
Regrouping the terms appropriately.
=
eiα

2
= 1− α
2!





4 α6
3 α5 α7
α
α


+ − + L + i α − + − +L and so...
4! 6!
3! 5! 7!






eiα = cos  α  + isin(α) Euler's Formula (α must be in radians)
Revised August 3, 2000
Steve Boast
Page 7