Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Complex Analysis → Complex Numbers ↓


Exponential Form


Complex numbers are a fundamental concept in mathematics that extends the idea of a one-dimensional number line to a two-dimensional complex plane. They are written in the form a + bi, where a and b are real numbers, and i is the imaginary unit, satisfying i 2 = -1.

Traditional representation

Typically, complex numbers are represented in their standard form, where they are expressed as the sum of the real part a and the imaginary part bi. This is known as the rectangular form. However, there is another compelling and more practical representation known as the exponential form, which takes advantage of Euler's formula.

Understanding Euler's formula

Euler's formula is a fundamental equation in complex analysis that establishes a deep connection between complex exponentials and trigonometric functions. It is stated as follows:

e iθ = cos(θ) + i*sin(θ)

Example 1:

For θ = π, Euler's formula gives:

e iπ = cos(π) + i*sin(π) = -1 + i*0 = -1

Hence, e iπ + 1 = 0, known as Euler's identity, is often celebrated for its beauty as it connects five fundamental mathematical constants: 0, 1, π, e, and i.

Exponential form of a complex number

Using Euler's formula, any complex number can be expressed in its exponential form. Given a complex number, r(cos θ + i sin θ) in polar coordinates, it can be rewritten using Euler's formula as:

z = re iθ

Here, r is the magnitude (or modulus) of the complex number, which is the distance from the origin to the point in the complex plane. θ is the argument (or angle), measured from the positive real axis to the line representing the complex number.

Example 2: Exponential transformation

Suppose we have a complex number z = 1 + i√3. We need to convert it to its exponential form:

  1. Calculate the magnitude r:
    r = √(1 2 + (√3) 2) = √4 = 2
  2. Calculate the argument θ:
    θ = atan2(√3, 1) = π/3
  3. Thus, the exponential form is:
    z = 2e iπ/3

Advantages of using exponential form

One of the main advantages of the exponential form of complex numbers is that it greatly simplifies operations such as multiplication, division, and root extraction. In exponential form, complex number multiplication becomes simple: just multiply the magnitudes and add the angles.

Operations with exponential form

Suppose we have two complex numbers in exponential form: z1 = r1e iθ1 and z2 = r2e iθ2. Here are the basic operations:

Multiplication of complex numbers

z1 * z2 = (r1 * r2)e i(θ1 + θ2)

Example 3: Multiplication

Let z1 = 3e iπ/4 and z2 = 2e iπ/6.

z1 * z2 = (3 * 2)e i(π/4 + π/6) = 6e i(5π/12)

Division of complex numbers

z1 / z2 = (r1 / r2)e i(θ1 - θ2)

Example 4: Division

Divide by z1 = 4e iπ/3 z2 = 2e iπ/6.

z1 / z2 = (4 / 2)e i(π/3 - π/6) = 2e iπ/6

Finding powers

For powers we use the following formula:

z n = (r n)e inθ

Example 5: Powers

Let z = 2e iπ/4. Find z 3.

z 3 = (2 3)e i(3π/4) = 8e i3π/4

Finding the roots

n root of the complex number z = re iθ can be found as follows:

z k = r 1/n e i(θ+2kπ)/n for k = 0, 1, ..., n-1

Example 6: Roots

Find the cube root of z = 8e iπ.

z k = 8 1/3 e i(π+2kπ)/3 for k = 0, 1, 2

This process gives the roots 2e iπ/3, 2e i(π+2π)/3, and 2e i(π+4π)/3.

Viewing complex numbers in exponential form

The exponential form of a complex number can be viewed as a rotation around the origin of the complex plane. In the SVG below, assume a circle of radius r centered at the origin, and the angle θ represents the rotation about the positive x-axis. Each complex number can be viewed as a point on this circle.

Jade θ R

Conclusion

The exponential form of complex numbers is not just a different way to write complex numbers; it provides insight and simplifications, especially in complex calculus, signal processing, and Fourier analysis. This form integrates seamlessly with the concepts of angles and rotations, making it invaluable in both theoretical and practical applications in various fields of mathematics and engineering.


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Exponential Form

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