Mathematics Database Programming Web Design Price List     Complex analysis studies the most unexpected, surprising, even paradoxical ideas in mathematics. The familiar rules of math of real numbers may break down when applied to complex numbers. Free Lessons Lesson 1 Complex numbers In this lesson: Forms and representations of the complex numbers; Modulus and arguments; Principal value of the argument. Lesson 2 Trigonometric and algebraic form conversion In this lesson: Complex numbers forms conversion; Examples of the conversion. Lesson 3 The algebra of complex numbers In this lesson: Arithmetic operations with complex numbers; Properties of the complex numbers; Geometric interpretation of addition & subtraction. Lesson 4 Geometric interpretation of multiplication In this lesson: The modulus and argument of the product; Multiplication of complex numbers as stretching - squeezing and rotation; Multiplying a complex number by imaginary unit i and by powers of i. Lesson 5 Division of the complex numbers In this lesson: Definition and notation of conjugates and reciprocals; Division as multiplication and reciprocation. Lesson 6 Powers and roots of complex numbers In this lesson: De Moire's theorem; Powers of complex numbers; n-th root of complex numbers. Lesson 7 Complex Exponential Function and Complex Logarithm Function In this lesson: Definition and notation; Complex logarithm function is a multi-valued function; Principal branch of the logarithm. Lesson 8 Complex Trigonometric Functions and Complex Inverse Trigonometric Functions In this lesson: A difference between the real and complex trigonometric functions; Relationship to exponential function; Identities; Derivatives and Indefinite integrals of inverse trigonometric functions. Lesson 9 Complex Hyperbolic Functions and Inverse Hyperbolic Functions In this lesson: The notations; Definitions; Derivatives and Indefinite integrals of inverse hyperbolic functions. Lesson 10 Complex Power Function In this lesson: Raising a complex number to a complex power; Derivatives and Indefinite integral of complex power function. Lesson 11 Complex Rational Functions In this lesson: Definition of the rational function; Möbius transformations; Fractional-linear function; Zhukovskii function. The algebra of complex numbers Complex Analysis. FreeTutorial

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5.1 Arithmetic operation with complex numbers

z1 = x1+ y1i,
z
2 = x2+ y2i.
Assuming that the ordinary rules of arithmetic apply to complex numbers we can find:

z1 + z2 = (x1+y1i) + (x2+y2i) = (x1+x2) + i(y1+y2)                (1.10)

Subtraction
z1 - z2 = (x1+y1i) - (x2+y2i) = (x1-x2) + i(y1-y2)

Multiplication
z1z2 = (x1+y1i)(x2+y2i) = x1x2 + y1x2i + x1y2i +y1y2i2
z1z2 = (x1x2 - y1y2) + (y1x2 + x1y2)i       (1.11)

Example:
z1 = 0 + 1i = i;
z
2= 0 + 1i = i.
z1z2 = i2 = (0+1i)(0+1i) = (0 -1) + (0+0)i = -1

Division
z1/z2 = (x1+y1i)/(x2+y2i) is a complex number, provided that z2 = x2+ y2i 0. (1.12)
x22+ y22 0.
To get the equation (1.12), we force the complex denominator to be real by multiplying both numerator and denominator by the number x2 -y2i. =

= .             (1.13)

After multiplication (x1+y1i)(x2 -y2i) we get the formula (1.12).

Example 13:
There are two numbers:
z
1 = 5 - 3i,
z
2 = 1 + 4i.
z
1 + z2 = (5 - 3i) + (1 + 4i) = (5 + 1) + (-3 + 4)i = 6 + i
Subtraction:
z
1 - z2 = (5 - 3i) - (1 + 4i) = (5 - 1) + (-3 - 4)i = 4 - 7i
Multiplication:
z
1z2 = (5 - 3i)(1 + 4i) = 5 - 3i + 20i - 12i2 = 17 + 17i
Division:
z1/z2 = (5 - 3i)/(1 + 4i) = {(-12 +5) + i(-3 - 20)}/(16 + 1) = - 7/17 - i(23/17)

Example 14:
z1 = 1+ i,
z
2 = 1- i.
z1/z2 = (1 + i)/(1- i) = (1 + i)2/{(1- i)(1+ i)} = (1 + 2i + i2)/(1 - i2) = 2i/2 = i.

5.2 Properties of the complex numbers

z1 + z2 = z2 + z1 Proof

2. Commutative law for multiplication:
z1z2 = z2z1 Proof

z1 + (z2 + z3) = (z1 + z2) + z3

4. Associative law for multiplication:
z1(z2z3) = (z1z2)z3

5. Multiplication is distributive with respect to addition:
z1(z2 + z3) = z1z2 + z1z3

6. The product of two complex numbers is zero if and only if at least one of the factors is zero.

Any complex number z has a unique negative –z such that z + (–z) = 0. If z = x + yi, the negative –z = – x yi.

8. Multiplicative Inverses:
Any nonzero complex number z = x + yi has a unique inverse 1/z such that z(1/z) = 1.
The number 1/z is called the reciprocal of the complex number z.
1/z = .                        (1.14)

There is a complex number w such that z + w = z for all complex numbers z. The number w is the ordered pair (0, 0).

10. Multiplicative Identity.
There is a complex number such that z = z for all complex numbers z. The ordered pair (1, 0) = 1 + 0i is the unique complex number with this property.

5.2.1 Proof of the properties of the complex numbers

None of these properties is difficult to prove. Most of the proofs use the corresponding facts in the real number system:
Real numbers are commutative under addition
x
+ y = y + x.
Real numbers are commutative under multiplication
x·y = y·x.

1. Proof of the commutative law for addition
Let us prove that z1 + z2 = z2 + z1.
z1 = x1 + iy1; z2 = x2 + iy2.

By definition of addition of complex numbers (1.10) z1 + z2 = (x1 + x2) + i(y1 + y2).
x1, x2, y1, y2 are all real and it doesn't matter what order real numbers are added up in.
By the commutative law for addition for real numbers
(x1 + x2) = (x2 + x1) and (y1 + y2) = (y2 + y1).

z1 + z2 = (x1 + x2) + i(y1 + y2) =
(x2 + x1) + i(y2 + y1)
= z2 + z1

2. Proof of the commutative law for multiplication
Let us prove that z1z2 = z2z1.
z
1 = x1 + iy1; z2 = x2 + iy2.

By definition of multiplication of complex numbers (1.11) z1z2 = (x1x2 - y1y2) + (y1x2 + x1y2)i
and z2z1 = (x2x1 - y2y1) + (y2x1 + x2y1)i.

x1, x2, y1, y2 are all real. By the commutative law for multiplication for real numbers
x
1x2 = x2x1
and y1y2 = y2y1.
It means that x1x2 - y1y2 = x2x1 - y2y1
and y1x2 + x1y2 = y2x1 + x2y1.
Such way z1z2 = z2z1.

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