Why are some coefficients negative when expanding a function in Legendre?

In summary, the Legendre function is a mathematical function used to represent and expand symmetric functions, named after Adrien-Marie Legendre. To expand a function in Legendre, it must first be expressed as a series of Legendre polynomials, which can be done using the Rodrigues' formula or generating function. This method has many applications in physics and engineering, such as solving partial differential equations and approximating complicated functions. Legendre functions have advantages such as orthogonality and a recursive relationship, but they also have limitations, such as only being applicable to symmetric functions and potentially not being the most efficient method of expansion for certain functions.
  • #1
chenaiyy
1
0
I get a series of Legendre expansion coefficients for a function. Then I compute the value of the function via expansion coefficients. As I want to whether the code is right or not, finally I expand the function in Legendre again. the result is almost same as before, but some coefficients is negative which are not exist before. I want to know why ?
The code are all fortran.
Thanks a lot!
 
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  • #2
You'll have to show the code!
 

FAQ: Why are some coefficients negative when expanding a function in Legendre?

What is the Legendre function?

The Legendre function is a mathematical function used to represent and expand functions that are symmetric about a point. It is named after the French mathematician Adrien-Marie Legendre who first introduced it in 1782.

How do you expand a function in Legendre?

To expand a function in Legendre, you first need to express the function as a series of Legendre polynomials. This can be done by using the Rodrigues' formula or by using the generating function of Legendre polynomials. Once the function is expressed in terms of Legendre polynomials, it can be expanded using the orthogonality property of Legendre polynomials.

What is the significance of expanding a function in Legendre?

Expanding a function in Legendre has several applications in physics and engineering. It is used to solve partial differential equations, to represent symmetric functions, and to approximate complicated functions with simpler ones. It is also used in the study of quantum mechanics and spherical harmonics.

What are the advantages of using Legendre functions for expansion?

Legendre functions have several advantages over other types of functions for expansion. They are orthogonal, meaning that their inner product is zero, which simplifies the expansion process. They also have a recursive relationship, making it easier to calculate higher-order Legendre polynomials. Additionally, they have a wide range of applications and are well-studied and understood in mathematics.

Are there any limitations to expanding a function in Legendre?

While Legendre functions have many advantages, they also have some limitations. They are only applicable for functions that are symmetric about a point, so they cannot be used for asymmetric functions. Additionally, they may not be the most efficient or accurate method of expansion for certain types of functions. In these cases, other types of functions, such as Chebyshev or Fourier, may be more suitable for expansion.

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