Special Functions

In the realm of mathematics, we encounter a vast array of functions, from the simple linear functions to the intricate complexities of calculus. Among these, a special class of functions emerges, known as special functions. These are not your everyday polynomials or trigonometric functions; they arise naturally in various branches of mathematics, physics, and engineering, often as solutions to differential equations or integrals that defy elementary solutions.

Key Characteristics:

  • Non-Elementary: Special functions cannot be expressed in terms of simple combinations of elementary functions like polynomials, exponentials, logarithms, or trigonometric functions.

  • Ubiquitous in Applications: They appear frequently in diverse fields:

  • Physics: Describing phenomena in quantum mechanics, electromagnetism, and fluid dynamics.

  • Engineering: Solving problems in signal processing, control theory, and heat transfer.

  • Mathematics: Arising in number theory, combinatorics, and complex analysis.

Examples of Special Functions:

  • Gamma Function: A generalization of the factorial function to real and complex numbers.

  • Bessel Functions: Solutions to Bessel’s differential equation, crucial in problems involving cylindrical symmetry (e.g., vibrations of drums, wave propagation in cylindrical structures).

  • Legendre Polynomials: Solutions to Legendre’s differential equation, fundamental in potential theory and spherical harmonics.

  • Error Function: Describes the probability of finding a normally distributed random variable within a certain range.

  • Elliptic Integrals: Arise in the calculation of arc lengths of ellipses and other geometric problems.

Bessel Functions

Bessel functions are a family of solutions to Bessel’s differential equation, which appears in many physical problems involving cylindrical or spherical symmetry. They are named after the German mathematician Friedrich Wilhelm Bessel, who first studied them in the early 19th century.

Bessel’s Differential Equation

The general form of Bessel’s differential equation is:

\[x^2 \frac{d^2y}{dx^2} + x\frac{dy}{dx} + (x^2 - n^2)y = 0\]

where \(n\) is a parameter that determines the order of the Bessel function.

Types of Bessel Functions

  1. Bessel Functions of the First Kind \((J_n(x))\) These functions are denoted by \((J_n(x))\) and are solutions to Bessel’s differential equation that are finite at the origin (for non-negative integer orders). They are commonly used in problems involving wave propagation, static potentials and flow in porous media.

    \[J_n(x) = \sum_{m = 0}^{\infty} \frac{(-1)^m}{m!\Gamma(m+n+1)}\left(\frac{x}{2}\right)^{2m + n}\]

  2. Bessel Functions of the Second Kind \((Y_n(x))\) These functions are denoted by \((Y_n(x))\), are also solutions to Bessel’s differential equation but have a singularity at the origin. They are often used in conjunction with \((J_n(x))\) to form a complete set of solutions.

    \[Y_n(x) = \frac{J_n(x)\cos(n\pi) - J_{-n}(x)}{\sin(n\pi)}\]

  3. Modified Bessel Functions ( \(I_n(x)\) and \(K_n(x)\) ): These functions are solutions to the modified Bessel’s differential equation, which is obtained by replacing \(x\) with \(ix\) in the original equation. They are used in problems involving heat conduction and diffusion.

    \[I_n(x) = \sum_{m = 0}^{\infty} \frac{1}{m!\Gamma(m+n+1)}\left(\frac{x}{2}\right)^{2m + n}\]
    \[K_n(x) = \frac{\pi}{2}\frac{I_{-n}(x) - I_n(x)}{\sin(n\pi)}\]
// import libraries
using System;
using SepalSolver;
using static SepalSolver.Math;

ColVec x = Linspace(0, 10);
Indexer Z = new(0, 8);
Matrix J = Z.Select(z => BesselJ(z, x)).ToList();
Plot(x, J, Linewidth: 2);
Axis([0, 10, -0.5, 1]);
Title("BesselJ Functions");
Legend(Z.Select(z => z.ToString()), Alignment.UpperRight);
SaveAs("BesselJ-Functions.png");

Output:

BesselJ-Functions.png 
ColVec x = Linspace(1, 10);
Indexer Z = new(0, 8);
Matrix Y = Z.Select(z => BesselY(z, x)).ToList();
Plot(x, Y, Linewidth: 2);
Title("BesselY Functions");
Axis([1, 10, -5, 1]);
Legend(Z.Select(z => z.ToString()), Alignment.UpperRight);
SaveAs("BesselY-Functions.png");

Output:

BesselY-Functions.png

Legendre Functions

Legendre polynomials are a set of orthogonal polynomials that arise in solving certain types of differential equations, particularly in physics and engineering. They are named after the French mathematician Adrien-Marie Legendre.

The Legendre polynomials \(P_n(x)\) are solutions to Legendre’s differential equation:

\[(1 - x^2) \frac{d^2y}{dx^2} - 2x\frac{dy}{dx} + n(n+1)y = 0\]

where \(n\) is a non-negative integer.

Some key properties of Legendre polynomials include:

  1. Orthogonality: They are orthogonal with respect to the weight function \(w(x) = 1\) on the interval \([-1,1]\).

  2. Normalization: \(P_n(1) = 1\) for all \(n\)

  3. Recurrence Relation: They satisfy the recurrence relation:

    \[(n+1)P_{n+1}(x) = (2n + 1)xP_{n}(x) - nP_{n-1}(x)\]

Types of Legendre polynomials

Legendre polynomials of the First Kind \((P_n(x))\)

\[P_n(z) = \frac{1}{2^n}\sum_{k=0}^{\lfloor \frac{n}{2} \rfloor} \frac{(-1)^k(2n-2k)!}{k!(n-k)!(n - 2k)!}x^{n-2k}\]
// import libraries
using System;
using SepalSolver;
using static SepalSolver.Math;

ColVec x = Linspace(-1, 1);
Indexer Z = new(0, 5);
Matrix P = Z.Select(z => Legendre(z, x)).ToList();
Plot(x, P, Linewidth: 2);
Title("LegendreP Functions");
Legend(Z.Select(z => z.ToString()), Alignment.UpperCenter);
SaveAs("LegendreP-Functions.png");

Output:

LegendreP-Functions.png

Legendre polynomials of the Second Kind \((Q_n(x))\)

\[Q_n(z) = \frac{1}{2}P_n(x)\ln\left(\frac{1+x}{1-x}\right) + \sum_{k=1}^{\lfloor \frac{n+1}{2} \rfloor} \frac{2n - 4k + 3}{(2k - 1)(n - k + 1)}P_{n - 2k + 1}(x)\]
// import libraries
using System;
using SepalSolver;
using static SepalSolver.Math;

ColVec x = Linspace(-0.95, 0.95);
Indexer Z = new(0, 5);
Matrix Q = Z.Select(z => LegendreQ(z, x)).ToList();
Plot(x, Q, Linewidth: 2);
Title("LegendreQ Functions");
Legend(Z.Select(z => z.ToString()), Alignment.UpperCenter);
SaveAs("LegendreQ-Functions.png");

Output:

LegendreQ-Functions.png

Chebyshev polynomials

Chebyshev polynomials are a sequence of orthogonal polynomials that are widely used in numerical analysis, approximation theory, and other areas of mathematics. There are two main types of Chebyshev polynomials: those of the first kind, denoted as \(T_n(x)\) and those of the second kind, denoted as \(U_n(x)\). ​

.

Types of Chebyshev polynomials

ChebyshevT polynomials of the First Kind \((T_n(x))\)

\[\begin{split}\begin{array}{rcl} T_0(x) &=& 1 \\ T_1(x) &=& x \\ T_{n+1}(x) &=& 2xT_n(x) - T_{n-1}(x) ~\text{for}~ n \geq 1 \end{array}\end{split}\]
// import libraries
using System;
using SepalSolver;
using static SepalSolver.Math;

ColVec x = Linspace(-1, 1);
Indexer Z = new(0, 5);
Matrix T = Z.Select(z => ChebyshevT(z, x)).ToList();
Plot(x, T, Linewidth: 2);
Title("ChebyshevT Polynomial Functions");
Legend(Z.Select(z => z.ToString()), Alignment.UpperCenter);
SaveAs("ChebyshevT-Polynomial-Functions.png");

Output:

ChebyshevT-Polynomial-Functions.png

ChebyshevU polynomials of the Second Kind \((U_n(x))\)

\[\begin{split}\begin{array}{rcl} U_0(x) &=& 1 \\ U_1(x) &=& 2x \\ U_{n+1}(x) &=& 2xU_n(x) - U_{n-1}(x) ~\text{for}~ n \geq 1 \end{array}\end{split}\]
// import libraries
using System;
using SepalSolver;
using static SepalSolver.Math;

ColVec x = Linspace(-1, 1);
Indexer Z = new(0, 5);
Matrix T = Z.Select(z => ChebyshevU(z, x)).ToList();
Plot(x, T, Linewidth: 2);
Title("ChebyshevU Polynomial Functions");
Legend(Z.Select(z => z.ToString()), Alignment.UpperCenter);
SaveAs("ChebyshevU-Polynomial-Functions.png");

Output:

ChebyshevU-Polynomial-Functions.png

Laguerre Polynomial

Laguerre polynomials are a sequence of orthogonal polynomials named after the French mathematician Edmond Laguerre. These polynomials are solutions to the Laguerre differential equation:

\[x \cfrac{d^2y}{dx^2} + (1 - x)\cfrac{dy}{dx} + ny = 0\]

where \(n\) is a non-negative integer. The Laguerre polynomials are denoted by \(L_n(x)\) and have several important properties and applications.

It can be generated by the following recurrent relation

\[\begin{split}\begin{array}{rcl} L_0(x) &=& 1 \\ L_1(x) &=& 1 - x \\ (n + 1)L_{n+1}(x) &=& (2n + 1 - x)L_n(x) - nL_{n-1}(x) ~\text{for}~ n \geq 1 \end{array}\end{split}\]
// import libraries
using System;
using SepalSolver;
using static SepalSolver.Math;

ColVec x = Linspace(-2, 10);
Indexer Z = new(1, 5);
Matrix P = Z.Select(z => Laguerre(z, x)).ToList();
Plot(x, P, Linewidth: 2);
Title("Laguerre Polynomial Functions");
Axis([-2, 10, -10, 10]);
Legend(Z.Select(z => z.ToString()), Alignment.UpperCenter);
SaveAs("Laguerre-Polynomial-Functions.png");

Output:

Laguerre-Polynomial-Functions.png

Hermite Polynomials

Hermite polynomials are a classical sequence of orthogonal polynomials that arise in various fields of mathematics and physics. Named after the French mathematician Charles Hermite, these polynomials are particularly significant in probability theory, combinatorics, and quantum mechanics.

\[x \cfrac{d^2y}{dx^2} - 2x\cfrac{dy}{dx} + 2ny = 0\]

where \(n\) is a non-negative integer. The Hermite polynomials are denoted by \(H_n(x)\) and have several important properties and applications.

It can be generated by the following recurrent relation

\[\begin{split}\begin{array}{rcl} H_0(x) &=& 1 \\ H_1(x) &=& 2x \\ H_{n+1}(x) &=& 2x H_n(x) - 2n H_{n-1}(x) ~\text{for}~ n \geq 1 \end{array}\end{split}\]
// import libraries
using System;
using SepalSolver;
using static SepalSolver.Math;

ColVec x = Linspace(-2, 2);
Indexer Z = new(0, 5);
Matrix T = Z.Select(z => Hermite(z, x)).ToList();
Plot(x, T, Linewidth: 2);
Title("HermiteH Polynomial Functions");
Axis([-2, 2, -30, 30]);
Legend(Z.Select(z => z.ToString()), Alignment.UpperCenter);
SaveAs("HermiteH-Polynomial-Functions.png");

Output:

HermiteH-Polynomial-Functions.png