# Convergence of expansion of Legendre generating function.

• scorpion990
In summary, the Legendre functions are defined using a generating function, but the series only converges for certain values of x and t. When deriving recursion formulas, the assumption is usually made that |x| < 1 and |t| < 1, but this may not always be valid. The Legendre functions are typically expanded in terms of t, and the region of convergence can be determined using Cauchy's integral formula and a series test.
scorpion990
The Legendre functions may be defined in terms of a generating function: $$g(x,t) = \frac{1}{\sqrt{1-2xt+t^2}}$$
Of course, $$\frac{1}{\sqrt{1+x}} =\sum^{\infty}_{n=0} (\stackrel{-.5}{n})x^n$$.

However, this series doesn't converge for all x. It only converges if |x| < 1. In our case, $$|t^2 - 2xt|$$ would have to be less than 1.

In the derivation of many recursion formulas, powers of t are set equal to each other. However, this isn't valid for all values of t and x... How come this method of derivation is still valid? Any help/insight would be appreciated.

The interval of convergence can depend on the point at which you are taking the
Taylor expansion - and, for that matter, which variable you are doing the
expansion in. The one you choose above is just one of many ways to expand it, but not
the way the Legendre functions show up.
The Legendre functions (http://en.wikipedia.org/wiki/Legendre_polynomials) would
come from an expansion in your variable t. The assumption for certain applications would
be that |x| < or =1 , |t| < 1 , which does not check for your series (eg. x=-1, t=.999999999). [ I don't recall off-hand how the series extends for |t|> or + 1 .]

As an expansion in t (|t|<1) , you can quickly verify the region of convergence (going to the complex plane) by using Cauchy's integral formula for upper bounds on the Taylor coefficients followed by a series test.

## 1. What is the Legendre generating function?

The Legendre generating function is a mathematical function used in the study of special functions, particularly in the field of orthogonal polynomials. It is defined as G(x,t) = 1/(1-2xt+t2)1/2, where x and t are variables.

## 2. What is meant by "convergence of expansion" in the context of the Legendre generating function?

The convergence of expansion refers to the properties of the coefficients of the power series expansion of the Legendre generating function. Specifically, it is concerned with the conditions under which this expansion converges and the behavior of the coefficients for different values of the variables x and t.

## 3. What is the significance of the convergence of expansion of the Legendre generating function?

The convergence of expansion is important because it allows us to use the power series expansion of the Legendre generating function to approximate other functions. This is particularly useful in the field of numerical analysis, where approximations are often necessary due to the complexity of certain mathematical problems.

## 4. What are the conditions for the convergence of expansion of the Legendre generating function?

The convergence of expansion depends on the values of the variables x and t. For the expansion to converge, the absolute value of x must be less than 1 and the absolute value of t must be less than 1.

## 5. How is the convergence of expansion of the Legendre generating function related to the properties of orthogonal polynomials?

The Legendre generating function is closely related to orthogonal polynomials, which have the property that their coefficients can be determined from the coefficients of the power series expansion of the Legendre generating function. Therefore, the convergence of the expansion of this function is crucial in determining the properties and behavior of orthogonal polynomials.

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