Complex Numbers: Fundamental Concepts and Applications

    Introduction to Complex Numbers

    Complex numbers represent a fundamental extension of real numbers, enabling mathematical operations beyond the real number system. Denoted as ℂ, the complex number system finds extensive applications in mathematics, engineering, and physics.

    Key Definition

    A complex number takes the form a+bia + bi , where:

    • aa : real component
    • bb : imaginary component
    • ii : imaginary unit where i2=−1i^2 = -1

    Basic Operations with Complex Numbers

    Addition and Subtraction

    Addition: (a+bi)+(c+di)=(a+c)+(b+d)i(a+bi)+(c+di)=(a+c)+(b+d)i

    Subtraction: (a+bi)−(c+di)=(a−c)+(b−d)i(a+bi)-(c+di)=(a-c)+(b-d)i

    Multiplication and Division

    Multiplication: (a+bi)(c+di)=(ac−bd)+(ad+bc)i(a+bi)(c+di)=(ac-bd)+(ad+bc)i

    Division: a+bic+di=(ac+bd)+(bc−ad)ic2+d2\frac{a+bi}{c+di}=\frac{(ac+bd)+(bc-ad)i}{c^2+d^2}

    Properties and Forms

    Complex Conjugate

    For a complex number z=a+biz=a+bi , its conjugate is:

    z‾=a−bi\overline{z}=a-bi

    Modulus and Argument

    Modulus: ∣z∣=a2+b2|z|=\sqrt{a^2+b^2}

    Argument: arg⁡(z)=arctan⁡(ba)\arg(z)=\arctan(\frac{b}{a})

    Alternative Representations

    Polar Form

    z=r(cos⁡θ+isin⁡θ)z=r(\cos\theta+i\sin\theta)

    Exponential Form

    z=reiθz=re^{i\theta}

    Powers and Roots of Complex Numbers

    Powers of Complex Numbers

    For a complex number in exponential form z=reiθz=re^{i\theta} :

    zn=(reiθ)n=rneinθz^n=(re^{i\theta})^n=r^ne^{in\theta}

    where n is a positive integer

    Complex Roots

    The n -th root of a complex number has n distinct values:

    wk=r1nei(θ+2kπ)nw_k=r^{\frac{1}{n}}e^{\frac{i(\theta+2k\pi)}{n}}

    where k=0,1,2,…,n−1k = 0,1,2,\ldots,n-1

    Key Properties

    • Every complex number (except 0) has exactly n distinct n -th roots
    • The roots form a regular polygon in the complex plane
    • Each successive root is obtained by rotating through an angle of 2πn\frac{2\pi}{n}

    Advanced Applications

    De Moivre's Theorem

    (cos⁡θ+isin⁡θ)n=cos⁡(nθ)+isin⁡(nθ)(\cos\theta+i\sin\theta)^n=\cos(n\theta)+i\sin(n\theta)

    Complex Analysis Foundations

    Cauchy-Riemann Equations

    For a complex function f(z)=u(x,y)+iv(x,y)f(z)=u(x,y)+iv(x,y):

    ∂u∂x=∂v∂y\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y} and ∂u∂y=−∂v∂x\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}