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Complex functions Tutorial
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.

A geometric interpretation of multiplication

Complex Analysis. FreeTutorial

8.1 The modulus and argument of the product

8.2 Multiplication of complex numbers as stretching (squeezing) and rotation
      8.2.1 Multiplying a complex number by a real number
      8.2.2 Multiplying a complex number by imaginary unit i
      8.2.3 Multiplying a complex number by -i
      8.2.4 Multiplying a complex number by positive integer powers of i
      8.2.5 Multiplying a complex number by negative integer powers of i

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8.1 The modulus and argument of the product

Let us consider two complex numbers z1 and z2 in a polar form.
z1 = r1(cos+i sin),
z2 = r2(cos+i sin).
Their product can be written in the form
z1z2 =r1r2[(coscos - sinsin) +
+ i(sincos + cossin)].
By means of the addition theorems of the sine and cosine this expression can be simplified to
z1z2 = r1r2[(cos(+) + i(sin(+)]. (1.18)
The product z1z2 has the modulus r1r2 and the argument +.
Figure 1.15 illustrates the multiplication of the complex numbers. The length of the z1z2 vector equals the product of the lengths of z1 and z2.

r1 = |z1|, = Arg(z1),
r2 = |z2|, = Arg(z2).

|z1 z2| = |z1| |z2|,        (1.19)
Arg(z1z2) = Arg(z1) + Arg(z2).      (1.20)

The formula (1.20) means that the set of values of Arg(z1z2) is obtained by forming all possible sums of value of Arg(z1) and Arg(z2).

geometry of multiplication image


Figure 1.15 The geometry of multiplication.

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8.2 Multiplication of complex numbers as stretching (squeezing) and rotation

The geometric interpretation of multiplication of complex numbers z1z2 is stretching (or squeezing) and rotation of vectors in the plane. If you have two complex numbers z1 and z2 (z1 0, z2 0), you can draw a vector z1, multiply its length by the |z2|, and rotate the resulting vector counterclockwise through the angle Arg(z2). If |z2| > 0, we deal with stretching. If |z2| < 0, it is a case of squeezing.

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8.2.1 Multiplying a complex number by a real number

Let us multiply a complex number z = x + yi by the real number a.
(x + yi)a = xa + yai.
The multiplication of a complex number by the real number a, is a transformation which stretches the vector by a factor of a without rotation.
Let us consider two cases: a = 2, a = 1/2. Geometrically, when we double a complex number, we double the distance from the origin, to the point in the plane. When we multiply a complex number by 1/2, the result will be half way between the origin and z.
Multiplication by a > 0 is a transformation which stretches the complex plane C by a factor of a away from the origin.
Multiplication by a < 0 is a transformation which squeezes C toward 0.

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8.2.2 Multiplying a complex number by imaginary unit i

zi = (x + yi)i = - y + xi .
The point z in C is located x units to the right of the imaginary axis and y units above the real axis (if x > 0, y > 0).
The point zi is located y units to the left, and x units above.
Multiplying by i has rotated a point z 90° counterclockwise around the origin to the point zi. See Figure 1.16

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8.2.3 Multiplying a complex number by -i

z(-i) = (x + yi)(-i) = y - xi.
Multiplication by -i gives a 90° clockwise rotation around the origin or a 270° counterclockwise rotation around the origin.

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8.2.4 Multiplying a complex number by positive integer powers of i

Multiplying z by i means 90° counterclockwise rotation of a point z around the origin to the point zi.
Multiplying z by i2 means 180° counterclockwise rotation of a point z around the origin to the point zi2 or 90° counterclockwise rotation of a point zi around the origin to the point zi2.

Multiplying z by i3 means 270° counterclockwise rotation of a point z around the origin to the point zi3 or 90° counterclockwise rotation of a point zi2 around the origin to the point zi3.

Multiplying z by i4 means 360° counterclockwise rotation of a point z around the origin. The point z corresponds to z, zi4, zi8, and so on. The circle is repeated.

Figure 1.16 shows the multiplication by powers of i. The point z corresponds to z, zi4, zi8, zi12, Ö . The point zi corresponds to zi, zi5, zi9, zi13 and so on.

multiplication by i image

Figure 1.16 Multiplication by i

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8.2.5 Multiplying a complex number by negative integer powers of i

Multiplying z by i -1 means 90° clockwise rotation of a point z around the origin to the point z(i)-1.

Multiplying z by i -2 means 180° clockwise rotation of a point z around the origin to the point z(i)-2.
The point z(i)-2 also corresponds to 90° clockwise rotation of a point z(i)-1 around the origin.

Multiplying z by i -3 means 270° clockwise rotation of a point z around the origin to the point z(i)-3.
The point z(i)-3 also corresponds to 90° clockwise rotation of a point z(i)-2 around the origin.

Multiplying z by i -4 means 360° counterclockwise rotation of a point z around the origin. The point z corresponds to z, zi -4, zi -8, and so on. The circle is repeated.

Figure 1.17 shows the multiplication by negative powers of i.

multiplication by negative power of i image
Figure 1.17 Multiplication by negative powers of
i

by Tetyana Butler

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