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 → Mathematical Physics ↓


Special Functions


In the world of mathematical physics, special functions play an important role. These functions often appear as solutions to differential equations that arise in physics and engineering problems. They are called "special" because they have well-established properties, specific uses, and historical significance, and they help make the solution of complex problems more manageable.

Commonly viewed special functions

Some of the most commonly found special functions include:

  • Gamma function
  • Bessel functions
  • Legendre polynomials
  • Hermite polynomials
  • Laguerre polynomials
  • Hypergeometric function

Gamma function

Let's start with the gamma function, which is an extension of the factorial function to complex numbers. For any positive integer n, the factorial is given by:

n! = n × (n-1) × ... × 1

The gamma function, represented by Γ(n), extends this concept further. It is defined by the integral:

Γ(n) = ∫ 0 ∞ t n-1 e -t dt

This corresponds to the factorial for positive integers:

Γ(n) = (n-1)!

For non-integers, the gamma function provides a way to "extrapolate" the concept of factorials. This is useful in many areas of mathematics and physics.

Gamma function (conceptual)

Bessel functions

Bessel functions are solutions of Bessel's differential equation and are used in problems involving cylindrical symmetry, such as vibrating membranes or heat conduction in cylindrical objects. The general form of a Bessel function is:

x² y'' + xy' + (x² - n²)y = 0

Bessel functions of the first kind, J n (x), are normal, and they can be written using an infinite series:

J n (x) = (x/2) n Σ k=0 ∞ (-1) k / (k! Γ(n+k+1)) (x²/4) k
Bessel functions

Legendre polynomials

Legendre polynomials arise in solving the Laplace equation in spherical coordinates, often in problems of potential theory. The Legendre differential equation is:

(1 - x²)y'' - 2xy' + n(n+1)y = 0

The Legendre polynomial can be generated using the Rodrigues formula:

P n (x) = 1/(2 n n!) d n /dx n [(x² - 1) n ]
Legendre polynomials

Hermite polynomials

Hermite polynomials are used in probability theory, especially in the context of the Gaussian distribution, as well as in solutions of the quantum harmonic oscillator in quantum mechanics. They satisfy the Hermite difference equation:

y'' - 2xy' + 2ny = 0

These can be defined using generating functions or recursion relations, such as:

H n+1 (x) = 2xH n (x) - 2nH n-1 (x)
Hermite polynomials

Laguerre polynomials

Laguerre polynomials are solutions of Laguerre's differential equation, which one encounters, for example, in quantum mechanics when solving the radial part of the Schrödinger equation for the hydrogen atom:

xy'' + (1 - x)y' + ny = 0

These can be generated using the Rodriguez formula:

L n (x) = e x d n /dx n (e -x x n )
Laguerre polynomials

Hypergeometric function

The hypergeometric function is very general and generalizes many other special functions. It is represented using the following series:

F(a, b; c; x) = Σ n=0 ∞ (a) n (b) n / (c) n × (x n / n!)

where (q) n denotes the Pochhammer symbol, which is a rising factorial:

(q) n = q(q+1)(q+2)...(q+n-1)
Hypergeometric

Applications of special functions

Special functions have wide applications in various fields like quantum mechanics, electrical engineering, aerodynamics, etc. For example:

  • Bessel functions appear in static potentials, heat conduction, and wave propagation.
  • Legendre and related Legendre functions appear in gravitational field and atomic structure calculations.
  • Hermite polynomials are found in probability theory and quantum mechanical wave functions.
  • Laguerre polynomials are used in quantum chemistry and the radial solution of the hydrogen atom.
  • Gamma functions are used in complex analysis and number theory.
  • Hypergeometric functions generalize many other functions and are used in physics, including the field of atomic physics.

Conclusion

The study of special functions is a vast field, as it involves both a depth of mathematics and a breadth of applications in the physical sciences. Each special function has developed a rich history with specific application contexts and problems. Understanding these functions allows one to delve into advanced theoretical and applied physics problems with powerful analytical tools.


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