Download Polar and exponential forms

Introduction
Polar form
Exponential form
Applications
Euler’s formula
Test
POLAR AND EXPONENTIAL
FORMS
ALGEBRA 7
INU0114/514 (MATHS 1)
Dr Adrian Jannetta MIMA CMath FRAS
Polar and exponential forms
1 / 14
Adrian Jannetta
Introduction
Polar form
Exponential form
Applications
Euler’s formula
Test
Objectives
This presentation will cover the following:
• Argand diagrams and the complex plane
• Modulus and argument
• Conversion between Cartesian, polar and exponential form
• Logarithm of a complex number
This will prepare us for later topics such as De Moivre’s theorem.
Polar and exponential forms
2 / 14
Adrian Jannetta
Introduction
Polar form
Exponential form
Applications
Euler’s formula
Test
Argand diagram
We’ve previously seen that complex numbers of the form x + yi are
said to be in Cartesian form because they can be represented as
points (x, y) on a graph called an Argand diagram.
yi
4
z3
z1 = 4 + 3i
z2
z2 = 4i
z1
3
z3 = −2 + 3i
2
z4 = −4
1
z5 = 3 − 2i
z4
x
−4 −3 −2 −1
−1
1
2
3
4
z5
−2
This means we can examine complex numbers using the rules of
geometry, trigonometry and algebra.
Polar and exponential forms
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Adrian Jannetta
Introduction
Polar form
Exponential form
Applications
Euler’s formula
Test
Polar form of a complex number
Argand diagrams show us how to express complex numbers in polar
form.
yi
A complex number in polar form is
defined by its modulus (length)
and argument (angle).
z = x + yi
The modulus is the distance r of
the point from the origin.
r
The argument is an angle θ
θ
x
measured anticlockwise from the
positive
In polar form
x-axis.the complex number z = x + yi becomes
y
x
z = r cos θ + (r sin θ )i
The modulus r and argument θ is given by
y
p
r = x2 + y 2 and θ = tan−1
x
and θ must be in the correct quadrant.
Polar and exponential forms
4 / 14
Adrian Jannetta
Introduction
Polar form
Exponential form
Applications
Euler’s formula
Test
Cartesian to polar form
Express z = −4 + i in polar form. Give the argument, in radians to 3
S.F.
In this case we have |z| =
p
17 .
Since z is in the second quadrant then arg z = π + tan−1 (− 41 ) = 2.897
(radians).
Therefore
z=
p
17(cos 2.897 + i sin 2.897)
A quick way of representing the polar form is to use the r cis θ or
r∠θ notations.
p
p
For example z = 17 cis 2.897 or z = 17∠2.897.
Polar and exponential forms
5 / 14
Adrian Jannetta
Introduction
Polar form
Exponential form
Applications
Euler’s formula
Test
Polar to cartesian form
Express z = 10∠ 7π
6 in cartesian form.
In this case we have |z| = 10 and θ = 7π
6 .
Therefore
7π
z = 10(cos 7π
6 + i sin 6 )
p
= 10(− 23 + i(− 21 ))
p
∴ z = −5 3 − 5i
Polar and exponential forms
6 / 14
Adrian Jannetta
Introduction
Polar form
Exponential form
Applications
Euler’s formula
Test
Exponential form
Consider the Maclaurin series for the exponential function:
ex = 1 + x +
x2 x3 x4 x5
+
+
+ ...
2! 3! 4! 5!
Substitute x = iθ to get a series for eiθ
eiθ = 1 + iθ +
(iθ )2 (iθ )3 (iθ )4 (iθ )5
+
+
+
...
2!
3!
4!
5!
Expanding the brackets and recalling that i2 = −1, i3 = −i, etc:
eiθ
Polar and exponential forms
i2 θ 2 i3 θ 3 i4 θ 4 i5 θ 5
+
+
+
...
2!
3!
4!
5!
θ 2 iθ 3 θ 4 iθ 5
−
+
+
...
= 1 + iθ −
2!
3!
4!
5!
= 1 + iθ +
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Adrian Jannetta
Introduction
Polar form
Exponential form
Applications
Euler’s formula
Test
Grouping the real and imaginary terms we get:
θ2 θ4
θ3 θ5
iθ
e
=
1−
+
−... +i θ −
+
...
2!
4!
3!
5!
The terms within the first set of brackets are the terms for the
infinite series for cos θ .
The terms within the second set of brackets are the terms of the
infinite series for the function sin θ .
Therefore:
eiθ = cos θ + i sin θ
This relationship is known as Euler’s Identity.
If we multiply both sides of this by the modulus r we obtain a
slightly more general form:
reiθ = r(cos θ + i sin θ )
Polar and exponential forms
8 / 14
Adrian Jannetta
Introduction
Polar form
Exponential form
Applications
Euler’s formula
Test
Cartesian to polar and exponential form
p
Express z = 3 − i in polar and exponential form.
The modulus is
|z| =
qp
( 3)2 + (−1)2 = 2
The argument is in the fourth quadrant so that
−1
arg z = tan( p
) + 2π =
3
11π
6
The polar form of the number is
11π
z = 2(cos 11π
6 + i sin 6 )
and the exponential form is
z = 2ei
Polar and exponential forms
11π
6
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Adrian Jannetta
Introduction
Polar form
Exponential form
Applications
Euler’s formula
Test
Powers
Powers of complex numbers
Changing a complex number to exponential or polar form makes it easy to
evaluate powers of that number.
Evaluating a power
Evaluate (1 − i)4 .
We can write 1 − i in exponential form as
p 7π
1 − i = 2e 4 i
Raising both sides to the power of 4:
(1 − i)4
=
p 7π
( 2e 4 i )4 = 4e7πi
Changing back to cartesian form (via polar form):
(1 − i)4
4
∴ (1 − i)
=
4(cos 7π + i sin 7π)
=
4(−1)
=
−4
For higher powers (or nth roots) it may be necessary to use De Moivre’s theorem.
We’ll study this method at a later time.
Polar and exponential forms
10 / 14
Adrian Jannetta
Introduction
Polar form
Exponential form
Applications
Euler’s formula
Test
Logarithms
Logarithms of complex numbers
The function ln x for x ∈ R is defined on the domain x > 0, but we
can extend the domain to x ∈ C.
Complex logarithm
p
Evaluate ln(1 + 3i).
p
Express 1 + 3i in exponential form. Here we have r = 2 and
θ = 13 π.
Therefore
π
p
ln(1 + 3) = ln 2ei 3
Use log rules to simplify:
π
= ln 2 + ln ei 3
p
ln(1 + 3) = ln 2 + i π3
Polar and exponential forms
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Adrian Jannetta
Introduction
Polar form
Exponential form
Applications
Euler’s formula
Test
Logarithms
Multivalued functions
In the previous example the angle calculated was the principal value:
π
3.
We could also have chosen θ = 31 π + 2π = 37 π to represent the same
logarithm value. In general, the complex logarithm can take the values
specified by
p
n∈Z
ln(x + yi) = ln x2 + y 2 + arg(x + yi) + 2nπ,
Although the exponential (and polar) forms differ by multiples of 2π —
the Cartesian forms are all identical.
The complex logarithm is sometimes called “a multivalued function”. As
we saw in Semester 1, functions are supposed to have one input and one
output so this name is a misnomer!
We will return to this concept with De Moivre’s theorem in the final part
of the course.
Other examples of multivalued functions are square-roots, in which
every real number is associated with two roots.
Polar and exponential forms
12 / 14
Adrian Jannetta
Introduction
Polar form
Exponential form
Applications
Euler’s formula
Test
Finally...a beautiful equation!
Given the relationship
reiθ
= r(cos θ + i sin θ )
If we put r = 1 and θ = π into we obtain
(1)eiπ = eiπ (1)(cos π + i sin π)
⇒ eiπ = −1
Rearranging we can make this equation
eiπ + 1 = 0
This is called Euler’s Formula and it is often cited as being one of
the most beautiful discoveries in all of mathematics.
It relates, in a single equation, five of the most important numbers
in mathematics.
Polar and exponential forms
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Adrian Jannetta
Introduction
Polar form
Exponential form
Applications
Euler’s formula
Test
Test yourself...
Given the complex numbers
z1 = 5i
z2 = −5 − 12i
z3 = 10ei
3π
4
1 Find |z1 | and arg z1
2 Express z2 in polar and exponential form.
3 Express z3 in Cartesian form.
4 Calculate z2 z3 in polar form
5 Calculate ln z3
6 Calculate ln(−2)
Answers:
1 |z1 | = 5 and arg z1 =
π
2
2 z2 = 13(cos 4.318 + i sin 4.318) and
z = 13e4.318i
p
p
3 z3 = −5 2 + 5 2i
Polar and exponential forms
4 z2 z3 = (13)(10)ei(4.318+
3π
4 )
= 130e6.674i
5 ln z3 = ln 10 + 34 πi
6 −2 = 2eiπ . Therefore ln(−2) = ln 2 + πi
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Adrian Jannetta