Special Functions and Polynomials

In summary, special functions and polynomials are mathematical functions that have unique properties and are commonly used in various fields such as physics, engineering, and statistics. They cannot be expressed in terms of elementary functions, but can often be approximated by simpler functions. These functions have special properties such as orthogonality and recurrence relations, and are often related to each other through transformations and identities. Their versatile nature makes them essential tools in many mathematical and scientific contexts.
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Astronuc
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PF Member Careful pointed to the website of Gerardus 't Hooft, Dutch physicist and winner of 1999 Nobel Prize in Physics with Martinus J.G. Veltman. 't Hooft has a very interesting and useful website, which includes the following useful pdf file about 'Special Functions and Polynomials'.

http://www.phys.uu.nl/~thooft/lectures/specialfct.pdf

Includes:

Legendre polynomials
Associated Legendre polynomials
Bessel and Hankel functions
Spherical Bessel functions
Hermite polynomials
Laguerre polynomials
Associated Laguerre polynomials
Tschebyschev (Chebyshev) polynomials
 
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Thanks for sharing!
 

1. What are special functions and polynomials?

Special functions and polynomials are mathematical functions that have unique properties and play important roles in various areas of mathematics and physics. They are typically defined as functions that cannot be expressed in terms of elementary functions like polynomials, exponential or trigonometric functions. Examples of special functions include Bessel functions, Legendre polynomials, and gamma function.

2. How are special functions and polynomials used in real-world applications?

Special functions and polynomials are used in a wide range of fields, including physics, engineering, and statistics. For example, Bessel functions are used to describe oscillatory phenomena in physics, while Legendre polynomials are used in solving problems involving spherical symmetry. The gamma function is used in probability and statistics to calculate the areas under curves and to describe the distribution of data.

3. What are the properties of special functions and polynomials?

Special functions and polynomials have unique properties that make them useful in various mathematical and scientific contexts. Some common properties include orthogonality, recurrence relations, and asymptotic behavior. These properties allow for efficient computation and often reveal important relationships between different special functions and polynomials.

4. How are special functions and polynomials related to each other?

Special functions and polynomials are often related to each other through various transformations and identities. For example, Bessel functions can be expressed in terms of other special functions such as hypergeometric functions, and Legendre polynomials can be written as a special case of Jacobi polynomials. These relationships help to simplify calculations and expand the range of applications for special functions and polynomials.

5. Can special functions and polynomials be approximated by simpler functions?

While special functions and polynomials cannot be expressed in terms of elementary functions, they can often be approximated by simpler functions such as Taylor series or Chebyshev polynomials. These approximations can be useful in numerical calculations and can provide insight into the behavior of special functions and polynomials at different points. However, they may not accurately capture all of the properties and behaviors of the original special function or polynomial.

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