Complex Numbers

Contents

Define Complex number and its Geometrical representation 1

Describe De-moivre’s theorem 2

Calculate roots of Complex numbers by using De-moivre’s theorem 4

Expand sin(nA), cos (nA), and tan(nA) in terms of sin(A), cos(A), and tan(A) respectively 5

Describe Circular Functions 7

Describe Hyperbolic Functions 9

Explain the relationship between Circular and Hyperbolic Functions 10

Describe the Real and Imaginary parts of Circular and Hyperbolic Functions 12

Describe Logarithm of Complex quantities 14

Define Complex number and its Geometrical representation

A complex number is a number of the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1. In other words, a complex number is a number that can be written as a sum of a real part and an imaginary part.

The real part of a complex number a + bi is the real number a, and the imaginary part is the real number bi. The imaginary part is often denoted as Im(z) and the real part as Re(z).

Geometrically, a complex number can be represented as a point in a two-dimensional coordinate system, called the complex plane or the Argand plane. The horizontal axis represents the real part of the complex number, and the vertical axis represents the imaginary part.

The complex number a + bi can be represented as the point (a,b) in the complex plane. This representation is sometimes called the Cartesian or rectangular form of the complex number.

Another way to represent a complex number is in polar form, where the complex number is represented by its magnitude (or absolute value) and argument (or angle) with respect to the positive real axis. The magnitude of a complex number z = a + bi is given by |z| = √(a² + b²), and the argument of z is given by arg(z) = tan^(-1)(b/a), where the angle is measured in radians.

In the polar form, a complex number can be represented as (r, θ), where r is the magnitude of the complex number and θ is the argument. This representation is sometimes called the polar form or the exponential form of the complex number.

The geometrical representation of complex numbers allows us to perform operations such as addition, subtraction, multiplication, and division, as well as to visualize complex functions and their properties.

Example: Consider the complex number z = 3 + 4i. The geometric representation of this complex number in the complex plane.

To represent the complex number z = 3 + 4i in the complex plane, we first draw the horizontal and vertical axes, representing the real and imaginary parts, respectively. Then we locate the point (3, 4) in the plane, corresponding to the real and imaginary parts of the complex number. This point is shown below:

The length of the line from the origin to the point (3, 4) is the magnitude of the complex number, which can be calculated as |z| = √(3² + 4²) = √25 = 5. The angle between the positive real axis and the line connecting the origin and the point (3, 4) is the argument of the complex number, which can be calculated as arg(z) = tan^(-1)(4/3) = 0.93 radians or approximately 53.13 degrees.

Therefore, the geometric representation of the complex number z = 3 + 4i in the complex plane is a point located at a distance of 5 units from the origin, and at an angle of approximately 53.13 degrees counterclockwise from the positive real axis, as shown in the figure above.

Describe De-moivre’s theorem

This Learning Outcome requires an understanding of De-Moivre’s theorem and its applications. In this note, we will discuss De-Moivre’s theorem, its significance, and some examples of how it can be used to solve problems.

De-Moivre’s Theorem is a mathematical formula that provides a relationship between the exponential representation of complex numbers and the trigonometric representation of complex numbers. It is named after French mathematician Abraham De-Moivre, who developed this theorem in the 18th century.

De-Moivre’s Theorem states that for any complex number z = r(cos θ + i sin θ), where r is the magnitude of the complex number and θ is its argument, we can represent it in the exponential form as z = r e^(iθ).

Using this formula, we can derive a formula for the nth power of a complex number z as:

zⁿ = rⁿ (cos nθ + i sin nθ)

This formula is particularly useful for simplifying complex calculations involving powers of complex numbers. For example, consider the calculation of the cube of a complex number z = 1 + i. Using De-Moivre’s Theorem, we can find the cube of z as follows:

z³ = (1 + i)³

= 1³(cos 3π/4 + i sin 3π/4)

= 2(cos 3π/4 + i sin 3π/4)

Thus, we have simplified the calculation of the cube of a complex number to a simple trigonometric calculation.

De-Moivre’s Theorem also has applications in the field of electrical engineering. Electrical signals can be represented as complex numbers, and their properties can be analyzed using De-Moivre’s Theorem. For example, the calculation of the phase shift of an alternating current circuit can be simplified using this theorem.

In summary, De-Moivre’s Theorem is a powerful tool for simplifying complex calculations involving powers of complex numbers. Its applications extend beyond mathematics and are used in various fields, including engineering, physics, and computer science.

Calculate roots of Complex numbers by using De-moivre’s theorem

This Learning Outcome requires an understanding of how to calculate the roots of complex numbers using De-Moivre’s Theorem. In this note, we will discuss the process of calculating the roots of complex numbers and provide suitable examples.

De-Moivre’s Theorem states that for any complex number z = r(cos θ + i sin θ), we can represent it in exponential form as z = r e^(iθ). This theorem allows us to calculate the roots of complex numbers in a simple and efficient way. The nth roots of a complex number z can be calculated using the following formula:

z^(1/n) = [r^(1/n)] * [cos((θ+2kπ)/n) + i sin((θ+2kπ)/n)]

where k = 0, 1, 2, …, n-1.

This formula gives n different roots of the complex number z. The first root is given by k = 0, the second root by k = 1, and so on, up to the nth root given by k = n-1.

Let’s consider an example to illustrate how to use De-Moivre’s Theorem to calculate the roots of complex numbers. Suppose we want to find the square roots of the complex number z = 4 + 4i.

Step 1: Convert z to exponential form.

We have z = 4 + 4i = 4√2(cos π/4 + i sin π/4) = 4√2 e^(iπ/4)

Step 2: Apply the formula for the nth roots of z.

For n = 2, we have:

z^(1/2) = [4√2^(1/2)] * [cos((π/4+2kπ)/2) + i sin((π/4+2kπ)/2)]

where k = 0, 1.

Evaluating for k = 0, we get the first square root as:

z^(1/2) = [4√2^(1/2)] * [cos(π/8) + i sin(π/8)]

Evaluating for k = 1, we get the second square root as:

z^(1/2) = [4√2^(1/2)] * [cos(5π/8) + i sin(5π/8)]

Thus, we have calculated the two square roots of the complex number z using De-Moivre’s Theorem.

In conclusion, De-Moivre’s Theorem provides a simple and efficient way to calculate the roots of complex numbers. The formula for the nth roots of a complex number allows us to calculate n different roots, which can be useful in various applications in mathematics and engineering.

Expand sin(nA), cos (nA), and tan(nA) in terms of sin(A), cos(A), and tan(A) respectively

This Learning Outcome requires an understanding of how to expand trigonometric functions such as sin(nA), cos(nA), and tan(nA) in terms of sin(A), cos(A), and tan(A) respectively. In this note, we will discuss the process of expanding these trigonometric functions and provide suitable examples.

1. Expansion of sin(nA) in terms of sin(A) and cos(A):

Using the formula for the sine of the sum of two angles, we can expand sin(nA) as follows:

sin(nA) = sin[(n-1)A + A]

= sin[(n-1)A] cos(A) + cos[(n-1)A] sin(A)

By using the same formula repeatedly, we can expand sin(nA) in terms of sin(A) and cos(A) as a sum of terms of the form sin(kA)cos((n-k)A) for k = 0, 1, …, n.

For example, let’s expand sin(3A) in terms of sin(A) and cos(A):

sin(3A) = sin(2A + A)

= sin(2A) cos(A) + cos(2A) sin(A)

= [2sin(A)cos(A)]cos(A) + [cos²(A) – sin²(A)]sin(A)

= 2sin(A)cos²(A) + cos²(A)sin(A) – sin³(A)

1. Expansion of cos(nA) in terms of sin(A) and cos(A):

Using the formula for the cosine of the sum of two angles, we can expand cos(nA) as follows:

cos(nA) = cos[(n-1)A + A]

= cos[(n-1)A] cos(A) – sin[(n-1)A] sin(A)

By using the same formula repeatedly, we can expand cos(nA) in terms of sin(A) and cos(A) as a sum of terms of the form cos(kA)cos((n-k)A) for k = 0, 1, …, n.

For example, let’s expand cos(2A) in terms of sin(A) and cos(A):

cos(2A) = cos(A + A)

= cos(A)cos(A) – sin(A)sin(A)

= cos²(A) – sin²(A)

1. Expansion of tan(nA) in terms of tan(A):

Using the formula for the tangent of the sum of two angles, we can expand tan(nA) as follows:

tan(nA) = [tan((n-1)A + A)] / [1 – tan((n-1)A)tan(A)]

By using the same formula repeatedly, we can expand tan(nA) in terms of tan(A) as a fraction of polynomials in tan(A) of degree n-1 and degree n.

For example, let’s expand tan(2A) in terms of tan(A):

tan(2A) = [tan(A + A)] / [1 – tan(A)tan(A)]

= [tan(A) + tan(A)] / [1 – tan²(A)]

= (2tan(A)) / (1 – tan²(A))

In conclusion, expanding trigonometric functions such as sin(nA), cos(nA), and tan(nA) in terms of sin(A), cos(A), and tan(A) respectively can simplify the computation of these functions for large values of n. These expansions can be derived using the sum of angles formulas for trigonometric functions.

Describe Circular Functions

This Learning Outcome requires an understanding of circular functions. In this note, we will discuss what circular functions are and provide suitable examples.

Circular functions are functions that are defined in terms of the coordinates of a point on the unit circle. The unit circle is a circle centered at the origin with a radius of one unit. The coordinates of a point on the unit circle can be represented as (cos θ, sin θ), where θ is the angle between the positive x-axis and the line segment connecting the origin to the point.

There are three main circular functions: sine, cosine, and tangent. The sine function is defined as the y-coordinate of the point on the unit circle, while the cosine function is defined as the x-coordinate of the point on the unit circle. The tangent function is defined as the ratio of the sine function to the cosine function.

Formally, we define the sine function as follows:

sin θ = y

where (x, y) is a point on the unit circle with angle θ. Similarly, we define the cosine function as:

cos θ = x

where (x, y) is a point on the unit circle with angle θ. Finally, we define the tangent function as:

tan θ = y / x

where (x, y) is a point on the unit circle with angle θ.

Circular functions have a number of important properties that make them useful in various branches of mathematics and science. For example, the sine and cosine functions are periodic with a period of 2π. This means that their values repeat every 2π radians (or 360 degrees). The tangent function is also periodic, but with a period of π.

Circular functions can be used to model various physical phenomena, such as the motion of a pendulum or the oscillation of a spring. They are also used in many areas of mathematics, including trigonometry, calculus, and complex analysis.

For example, consider the function f(θ) = sin θ. The graph of this function is a wave that oscillates between -1 and 1, with a period of 2π. The graph of the function f(θ) = cos θ is also a wave that oscillates between -1 and 1, but it is shifted by a quarter of a period relative to the graph of the sine function.

In conclusion, circular functions are a fundamental concept in mathematics and science. They are used to model various physical phenomena and are an essential part of many branches of mathematics, including trigonometry, calculus, and complex analysis.

Describe Hyperbolic Functions

Hyperbolic functions are analogs of circular functions, and they are defined in terms of the hyperbola. A hyperbola is a curve that looks like two open curves that mirror each other, with a horizontal or vertical axis of symmetry. The hyperbolic functions are the hyperbolic sine (sinh), hyperbolic cosine (cosh), hyperbolic tangent (tanh), and their reciprocal functions.

The hyperbolic sine function, sinh(x), is defined as:

sinh(x) = (e^x – e^(-x)) / 2

Similarly, the hyperbolic cosine function, cosh(x), is defined as:

cosh(x) = (e^x + e^(-x)) / 2

Finally, the hyperbolic tangent function, tanh(x), is defined as:

tanh(x) = sinh(x) / cosh(x) = (e^x – e^(-x)) / (e^x + e^(-x))

Just like the circular functions, hyperbolic functions have a number of important properties that make them useful in various branches of mathematics and science. For example, the hyperbolic sine and cosine functions are related to the circular sine and cosine functions, respectively, by replacing the angle with a complex number. In particular, for any complex number z, we have:

sin(iz) = i*sinh(z)

cos(iz) = cosh(z)

Hyperbolic functions also have a variety of useful identities, including:

cosh²(x) – sinh²(x) = 1

sech²(x) = 1 – tanh²(x)

coth²(x) – 1 = csch²(x)

Hyperbolic functions are used in many areas of mathematics, including calculus, differential equations, and complex analysis. They also have important applications in physics, particularly in the study of electromagnetic fields and relativistic effects.

For example, consider the function f(x) = sinh(x). The graph of this function is a curve that grows exponentially as x increases. Similarly, the function g(x) = cosh(x) is a curve that also grows exponentially, but at a slightly slower rate than f(x). Finally, the function h(x) = tanh(x) is a curve that approaches 1 as x increases, and -1 as x decreases.

In conclusion, hyperbolic functions are a fundamental concept in mathematics and science. They are used to model various physical phenomena and are an essential part of many branches of mathematics, including calculus, differential equations, and complex analysis.

Explain the relationship between Circular and Hyperbolic Functions

This Learning Outcome requires an understanding of the relationship between circular and hyperbolic functions. In this note, we will discuss the relationship between these two types of functions and provide suitable examples.

Circular functions, such as sine and cosine, are defined in terms of the unit circle, while hyperbolic functions, such as sinh and cosh, are defined in terms of the hyperbola. Despite these different geometries, there is a strong relationship between the two families of functions.

The relationship between circular and hyperbolic functions can be understood by considering the complex plane. A complex number can be represented as a point in the complex plane, with the real part of the number represented on the horizontal axis, and the imaginary part represented on the vertical axis. The magnitude of the complex number is represented by its distance from the origin.

The circular functions can be defined in terms of the complex exponential function:

e^(ix) = cos(x) + i*sin(x)

where x is the angle in radians. This formula relates the circular functions to the complex exponential function. By analogy, we can define the hyperbolic functions in terms of the complex exponential function as well:

e^(x) = cosh(x) + sinh(x)

where x is a real number.

Notice that there is a difference between the arguments of the exponential function in these two formulas. In the circular case, the argument is a multiple of the angle, while in the hyperbolic case, it is the angle itself. This difference arises from the different geometries involved.

Another important relationship between circular and hyperbolic functions is given by the identities:

sin(ix) = i*sinh(x)

cos(ix) = cosh(x)

These formulas show that the hyperbolic functions can be obtained from the circular functions by replacing the argument with a multiple of the imaginary unit i.

The relationships between circular and hyperbolic functions have many applications in mathematics and science. For example, the hyperbolic functions can be used to describe the behavior of a spring that has been stretched or compressed beyond its equilibrium position. Similarly, circular functions are used to describe the motion of objects that move in circles, such as planets around the sun.

In conclusion, circular and hyperbolic functions are closely related, despite being defined in different geometries. The relationships between these families of functions have many important applications in mathematics and science.

Describe the Real and Imaginary parts of Circular and Hyperbolic Functions

This Learning Outcome requires an understanding of the real and imaginary parts of circular and hyperbolic functions. In this note, we will discuss the real and imaginary parts of circular and hyperbolic functions and provide suitable examples. Circular Functions:

The circular functions, such as sine and cosine, are defined in terms of the unit circle. These functions have real and imaginary parts that are related to the x and y coordinates of the point on the unit circle that corresponds to the angle. Specifically, we have:

cos(x) = Re(e^(ix))

sin(x) = Im(e^(ix))

where Re(z) denotes the real part of the complex number z, and Im(z) denotes the imaginary part of z. For example, if we take x = π/4, we have:

cos(π/4) = Re(e^(iπ/4)) = 1/√2

sin(π/4) = Im(e^(iπ/4)) = 1/√2

Thus, the real and imaginary parts of the circular functions are related to the cosine and sine of the angle, respectively.

Hyperbolic Functions:

The hyperbolic functions, such as sinh and cosh, are defined in terms of the hyperbola. These functions have real and imaginary parts that are related to the x and y coordinates of the point on the hyperbola that corresponds to the argument. Specifically, we have:

cosh(x) = Re(e^(x))

sinh(x) = Im(e^(x))

For example, if we take x = 1, we have:

cosh(1) = Re(e⁽¹⁾) = 1.5430806…

sinh(1) = Im(e⁽¹⁾) = 1.1752012…

Thus, the real and imaginary parts of the hyperbolic functions are related to the even and odd parts of the exponential function, respectively.

Relationship between Circular and Hyperbolic Functions:

The relationship between the real and imaginary parts of circular and hyperbolic functions can be seen by using the complex exponential function. Specifically, we have:

e^(ix) = cos(x) + i*sin(x)

e^(x) = cosh(x) + sinh(x)

By comparing the real and imaginary parts of these two equations, we can see that the circular functions correspond to the even and odd parts of the hyperbolic functions, respectively. Specifically, we have:

cos(x) = (cosh(ix) + cos(ix))/2

sin(x) = (cosh(ix) – cos(ix))/(2i)

Similarly, we can express the hyperbolic functions in terms of circular functions by using complex numbers. Specifically, we have:

cosh(x) = (e^(x) + e^(-x))/2

sinh(x) = (e^(x) – e^(-x))/2

In conclusion, the real and imaginary parts of circular and hyperbolic functions are related to the geometry of the unit circle and hyperbola, respectively. The relationship between the real and imaginary parts of these functions can be understood by using the complex exponential function.

Describe Logarithm of Complex quantities

This Learning Outcome requires an understanding of logarithms of complex quantities. In this note, we will discuss the logarithm of complex numbers and provide suitable examples.

Logarithms of Complex Quantities:

The logarithm of a complex number z = x + iy is defined as the complex number w = u + iv that satisfies the equation e^w = z. Specifically, we have:

w = ln(z) = ln|z| + i arg(z)

where |z| = √(x² + y²) is the magnitude of the complex number z, and arg(z) is the argument (or phase) of z, which is the angle between the positive x-axis and the line connecting the origin to z in the complex plane.

The argument of a complex number is not unique, since adding any integer multiple of 2π to the argument produces a new argument. Therefore, we must choose a branch of the logarithm function that assigns a unique value to the argument.

For example, let us find the logarithm of the complex number z = 1 + i. We have:

|z| = √(1² + 1²) = √2

arg(z) = arctan(1/1) = π/4

Therefore, we can write:

ln(z) = ln|z| + i arg(z) = ln(√2) + i (π/4)

Properties of Logarithms of Complex Quantities:

The properties of logarithms of complex numbers are similar to the properties of real logarithms. Specifically, we have:

ln(z₁ z₂) = ln(z₁) + ln(z₂)

ln(z₁ⁿ) = n ln(z₁)

where z₁ and z₂ are complex numbers, and n is an integer.

For example, let us find the logarithm of the product of two complex numbers, z₁ = 1 + i and z₂ = √3 – i. We have:

z₁ z₂ = (1 + i)(√3 – i) = √3 + (1 – √3)i

Therefore, we can write:

ln(z₁ z₂) = ln(√3 + (1 – √3)i) = ln|z₁ z₂| + i arg(z₁ z₂)

To find the argument of z₁ z₂, we can use the formula:

arg(z₁ z₂) = arg(z₁) + arg(z₂)

where the arguments are chosen to lie in the range (-π, π]. We have:

arg(z₁) = arctan(1/1) = π/4

arg(z₂) = arctan(-1/√3) = -π/6

Therefore, we can write:

arg(z₁ z₂) = π/4 – π/6 = π/12

Thus, we have:

ln(z₁ z₂) = ln|z₁ z₂| + i arg(z₁ z₂) = ln(2) + i (π/12)

In conclusion, the logarithm of a complex quantity is a complex number that satisfies the equation e^w = z. The properties of logarithms of complex quantities are similar to those of real logarithms.