Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate ↓


Complex Analysis


Complex analysis is a branch of mathematics that studies functions involving complex numbers. Complex numbers are numbers that have both a real part and an imaginary part, written as a + bi, where i is an imaginary unit with the property i2 = -1.

Understanding complex numbers

Let us first explain what complex numbers are. Traditionally, we are familiar with real numbers, which can be seen on the number line. However, to fully understand complex numbers, a plane is needed. This plane is called the complex plane or Argand plane.

In the complex plane, a complex number a + bi is represented as a point with coordinates (a, b). Here a is the real part and b is the imaginary part. The horizontal axis (also called the x-axis) is the real axis, and the vertical axis (y-axis) is the imaginary axis.

A+ Bye Real axis Imaginary axis

Operations with complex numbers

Add

To add two complex numbers, you just have to add their real parts and their imaginary parts separately. If you have two numbers (a + bi) and (c + di), their sum is given by:

(a + b) + (c + d) = (a + c) + (b + d)i

Multiplication

To multiply two complex numbers, use the distributive property just as in normal algebra, but remember that i2 = -1. For two numbers (a + bi) and (c + di), the product is:

(A + BI) * (C + DI) = AC + ADI + BCI + BDI2
= AC + (AD + BC)i + BD(-1)
= (AC – BD) + (AD + BC)i

Conjugate and modulus

Complex conjugates

The conjugate of a complex number a + bi is a - bi. It is reflected across the real axis in the complex plane.

Modulus

The modulus of a complex number a + bi, represented as |a + bi|, is the distance from the origin to the point (a, b) in the complex plane. It is given as:

|a + bi| = sqrt(a2 + b2)

The modulus can be viewed as the length of the line segment from the origin to the point represented by the complex number.

Functions of a complex variable

Complex analysis studies functions that take complex numbers as input and output. A function f of a complex variable z = x + yi can be written as f(z) = u(x, y) + vi(x, y), where u and v are functions of real variables x and y.

Analytical functions

A function is called analytic at a point if it is differentiable at that point and in some region around it. This is a stronger condition than differentiability in real analysis, and such functions have very nice properties. They are represented as power series, which makes them very powerful and useful.

Cauchy–Riemann equations

For a complex function f(z) = u(x, y) + vi(x, y) to be differentiable or analytic at a point, it must satisfy the Cauchy–Riemann equations:

∂u/∂x = ∂v/∂y
∂u/∂y = -∂v/∂x

Complex integration

Just as we integrate real functions, we can integrate complex functions as well. The integral of a complex function along a path in the complex plane is defined as:

∫ f(z) dz

Cauchy's integration theorem

A fundamental result in complex analysis is Cauchy's integral theorem. It states that for any closed path C in the domain where f is analytic:

∫_c f(z) dz = 0

This theorem means that if a function is analytic inside a closed curve, then the integral of that function around the curve is zero.

Cauchy's integral formula

Another important result is Cauchy's integral formula. It states that if f is analytic on and inside a simple closed curve C, and a is inside C, then:

f(a) = (1/2πi) ∫_C f(z)/(za) dz

This formula allows us to know the value of an analytic function inside a region, just by knowing its values on the boundary.

Series and remainder

Taylor series

If a function is analytic on and inside a circle, it can be expressed as a Taylor series, which is an infinite sum of terms. For example, f(z) around z = a is:

f(z) = f(a) + f'(a)(za) + (f''(a)/2!)(za)2 + ...

Laurent series

Laurent series can represent a wide class of functions and may include terms with negative degrees. For complex functions with singularities, Laurent series are a useful tool:

f(z) = ... + b_2/(za)2 + b_1/(za) + a_0 + a_1(za) + ...

Residue theorem

One of the most useful results in complex analysis is the residue theorem. It is useful for evaluating complex integrals. If f is analytic except at isolated singularities within C, then:

∫_C f(z) dz = 2πi * (sum of residues inside C)

Applications of complex analysis

Complex analysis has many applications, including engineering, physics, and other branches of mathematics. Here are some notable examples:

Fluid dynamics

Complex analysis is used to solve problems in fluid dynamics, especially those involving potential flow, where the use of complex potentials greatly simplifies the problem-solving process.

Electromagnetism

Techniques of complex analysis are used in solving electromagnetic problems. Concepts such as analytical continuation and the study of singularities are essential in these fields.

Signal processing

Both the Fourier transform and the Laplace transform are important tools in signal processing and control theory, and have their roots in complex analysis.

Complex analysis is not just a theoretical playground, but a practical tool that helps solve real-world problems, and provides insights that are not possible through real variables alone.

Conclusion

Complex analysis, with its rich theory and practical applications, provides many powerful techniques for tackling problems. Its ability to uniquely extend the reach of calculus into the realm of complex numbers gives it a vital role in both pure and applied mathematics. As you continue your mathematical journey, you will find complex analysis both interesting and essential.


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