The cis or exponential functions
Recall the unit circle definitions of the sine and cosine functions. Let
| cis(q) = cos(q)+isin(q)
= exp(iq) |
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of a purely imaginary argument. It is now easy to say how and why the
exponential property
cis(A)·cis(B) = cis(A+B)
or equivalenly
exp(iA)·exp(iB) = exp(i(A+B)) |
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follows immediately from the above add the angles, multiply the lengths definition
of complex multiplication. Hint: both factors have unit lengths.
The notation exp(iA) is employed because it is possible to define the
exponential exp(z) of a complex number z = a + ib where a and b are both
real. The definition for those of you who know about exponentials exp(a) of
real numbers a is as follows:
exp(a + ib) = exp(x) {cos(b) + i sin(b)).
For more details visit Calculus and Beyond.
Many trig identities follow from the above property (exponential property)
and the equality of two different ways to compute products of complex
numbers.
Cosine and Sine Addition Formulas
Preliminary Exercise: In the identity exp(iA)·exp(iB) =
exp(i(A+B)), express the left hand side in terms of the real and
imaginary parts of the factors.
Solution:
exp(iA)·exp(iB) = {cos(A) + i sin (A)} {cos(B) + i sin
(B))
= cos(A)cos(B)-sin(A)sin(B) + i{cos(A)sin(B) + sin(A)cos(B)}
exp(i(A+B)) = cos(A+B) + i sin(A+B).
The solution to the above exercise, implies the cosine and sine addition
formulas:
cos(A+B) = cos(A)cos(B)-sin(A)sin(B)
sin (A+B) = cos(A)sin(B) + sin(A)cos(B)
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Complex Numbers
with easy consequences of two ways
to multiply complex numbers in and between vectors & trig, etc
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[ Back ] [ Area Intro ] [ Next ]
The fundamental theorem of algebra and partial
fraction decomposition in calculus depend on complex numbers.
Easy Consequences
Vec & Cmplx No Applet
B2 C. Conjugates
B3 Pythagoras
B4 Distance
B5 Rt Triangle Similarity
B6 Trig., Functions
B7 Dot & Cross Products
B8 Cosine Law
B9 Exponential & cis fns
B10 Easy Trig Identities
B11 Set Viewpoint
Links: Interactive Maths
Hint: See the (newest) Complex
Number. Starter Lesson.
for a simple geometric introduction, then continue with easy consequence below.
The clearest geometric proof of the distributive law appears in the Euclidean-Geometry/Complex
No.s
folder.
First Earlier (Old) exposition of complex numbers follows
in Z1 to B1 below - read for review or revision .
First (Old) Complex No Intro
Distributive Law
A1 Add Poiints
A2 Polar Coords
A3 Polar Multiply
A4 Complex No.s
A5 Real Numbers
A6 Law of Signs
A7 Key Properties
B1 2nd Mult Method
C1 Unsigned Coords
C2 Signed Coords
C3 Set Codification
C4 More On Real No.s
D1 Arrow Navigation
D2 Sum of Motions
D3 Addition Method I
D4 Addition Method II
D5 Addition Method III
D6 Coordinate Addition
D7 1st Distributive Law
D8 2nd Distributive Law
D9 3rd Distributive Law
D1 to D6 after provide a review of vectors.
More on Complex Numbers:
Chapters in Volume 3::
19 Logs & Powers
19 Natural Log.
19 Exponential Fn.
20 What's Next
21 Add Vectors
22 Complex #'s
23 Complex #'s
23 Trig Identity
23 Proofs of.
24 Complex Logs etc
This further
Complex Number
Intro assumes the field properties
of real numbers in place of
deriving them geometrically
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