- #1
WARGREYMONKKTL
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some body who can explain for me the Legndre polynomials
Back Transformation for Legendre Polynomials
- Thread starter WARGREYMONKKTL
- Start date
In summary, the Legendre polynomials are a set of polynomials that are dependent on a continuous variable and have useful properties for dealing with spherical coordinates and potentials. The website, Google, can provide more information on them. If you have coefficients for different classes, you can use a back transformation to plot each class using the estimated coefficients.
- #2
quasar987
Science Advisor
Homework Helper
Gold Member
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What do you want to know and what do you already know about them
Surely, nobody will be willing to type the whole theory behind them so be more specific please
Surely, nobody will be willing to type the whole theory behind them so be more specific please
- #3
AlphaNumeric
- 290
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http://mathworld.wolfram.com/LegendrePolynomial.html
A set of polynomials dependent on continuous variable x and integers m (and sometimes n) which have useful orthogonality and spanning properties.
Known to have useful solutions when dealing with spherical coordinates and potentials. Want more info? There's a fantastic website I know called Google. It'll find anything for you ;)
A set of polynomials dependent on continuous variable x and integers m (and sometimes n) which have useful orthogonality and spanning properties.
Known to have useful solutions when dealing with spherical coordinates and potentials. Want more info? There's a fantastic website I know called Google. It'll find anything for you ;)
- #4
romant61
- 2
- 0
Legendre Polynomials
I need some help. I fitted a 7th order legendre polynomial and got the L0 to L7 coefficients for different classes. How can I get a back transformation in order to plot each class using the estimated coefficients?
Thanks to anybody.
Roberto.
I need some help. I fitted a 7th order legendre polynomial and got the L0 to L7 coefficients for different classes. How can I get a back transformation in order to plot each class using the estimated coefficients?
Thanks to anybody.
Roberto.
1. What are Legendre polynomials Pl(x)?
Legendre polynomials Pl(x) are a set of orthogonal polynomials that are named after French mathematician Adrien-Marie Legendre. They are used in mathematical applications, particularly in physics and engineering, to represent functions in terms of an infinite series.
2. How are Legendre polynomials Pl(x) calculated?
Legendre polynomials Pl(x) can be calculated using the recursive formula:
P0(x) = 1, P1(x) = x,
Pn+1(x) = ((2n+1)xPn(x) - nPn-1(x))/(n+1), where n is the degree of the polynomial.
3. What are the properties of Legendre polynomials Pl(x)?
Some important properties of Legendre polynomials Pl(x) include:
- They are orthogonal, meaning that the integral of their product over a specific interval is equal to zero.
- They are normalized, which means that the integral of their square over the same interval is equal to 1.
- They have real coefficients.
- They are symmetric or anti-symmetric, depending on the degree of the polynomial.
4. What are the applications of Legendre polynomials Pl(x)?
Legendre polynomials Pl(x) have various applications in mathematics, physics, and engineering. Some examples include:
- Representing physical phenomena such as electric potential and gravitational potential.
- Solving differential equations in fields like quantum mechanics and fluid dynamics.
- Approximating functions in numerical analysis.
- Generating random numbers in computer science.
5. Can Legendre polynomials Pl(x) be generalized to higher dimensions?
Yes, Legendre polynomials Pl(x) can be generalized to higher dimensions, such as 2D and 3D. These are known as Legendre functions and are used in fields like electromagnetism and quantum mechanics to represent functions in multiple variables.
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