Associated Legendre Polynomial of 1st and 2nd kind

In summary, Associated Legendre Polynomials of 1st and 2nd kind are mathematical functions used to represent spherical harmonics. They are defined by a series of terms containing both a polynomial and a trigonometric function. These polynomials are commonly used in physics and engineering to describe the behavior of electric or magnetic fields in spherical coordinates. The 1st kind polynomials are used for non-negative integer values of the degree, while the 2nd kind polynomials are used for negative integer values. Both types of polynomials have various properties and applications in different fields of mathematics and science.
  • #1
member 428835
Hi PF!

In MATLAB I'm trying to use associated Legendre polynomials of the 1st and second kind, widely regarded as ##P_i^j## and ##Q_i^j##, where ##j=0## reduces these to simply the Legendre polynomials of the 1st and second kind (not associated).

Does anyone here know if MATLAB has a built in function, or the most efficient way to build them? Also, I have Mathematica, and they are built in there.

Thanks!

Josh
 
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  • #3
Thanks!
 

What are Associated Legendre Polynomials of 1st and 2nd kind?

Associated Legendre Polynomials are mathematical functions used in spherical harmonics to describe the shape of a three-dimensional object. They are named after French mathematician Adrien-Marie Legendre.

What is the difference between 1st and 2nd kind Associated Legendre Polynomials?

The 1st kind Associated Legendre Polynomials are used to describe functions that are symmetric about the polar axis, while the 2nd kind Associated Legendre Polynomials are used for functions that are anti-symmetric about the polar axis.

How are Associated Legendre Polynomials of 1st and 2nd kind calculated?

The calculation of Associated Legendre Polynomials involves the use of the gamma function and the Legendre differential equation. The specific formula varies depending on the order and degree of the polynomial.

What are the applications of Associated Legendre Polynomials of 1st and 2nd kind?

Associated Legendre Polynomials have various applications in physics, engineering, and mathematics. They are used to describe the angular distribution of particles, electromagnetic fields, and gravitational fields. They are also used in the study of quantum mechanics and fluid dynamics.

Are there any real-world examples of Associated Legendre Polynomials of 1st and 2nd kind?

Yes, Associated Legendre Polynomials are used in many real-world scenarios, such as modeling the Earth's gravitational field for satellite missions, analyzing the structure of atomic orbitals, and calculating the magnetic field of planets. They are also used in image processing and pattern recognition algorithms.

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