User:Oh Isaac/R7-1

Problem 1: plot the local maximum and show equations
Report problem 5.10 from

Given
1.The hypergeometric series.

$$F(a,b;c;x)=\sum^\infty_{k=0}\frac{(a)_k(b)_k}{(c)_k}\frac{x^k}{k!}$$

2.Defination of local maximum.

A real-valued function f defined on a real line is said to have a local (or relative) maximum point at the point $$x_0$$, if there exists some$$\epsilon>0$$such that$$f(x_0)\geq f(x)$$ when$$|x-x_0|<\epsilon $$. The value of the function at this point is called maximum of the function. .

3.Determination of local maximum.

It is intuitively clear that the tangent line to the graph of a function at a local minimum or maximum must be horizontal, so the derivative at the point is 0, and the point is a critical point, which means $$f'(x_0)=0$$.

Find

 * 1) Use MATLAB to plot $$F(5,-10;1,x)$$ near $$ x=0 $$ to display the local maximum (or maxima) in this region.
 * 2) Show that :

Solution

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On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.
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Part 1
Display the local maximum (or maxima) near $$x=0$$.

Choose $$x\in [-10,10] $$as the neighborhood of $$x $$near $$x_0=0$$. Get the data using function of hypergeom([a,b],c,d) in MATLAB.

In order to get the local maximum of $$F(a,b;c,x)$$, we can find the points of the data whose differential value is 0 and make further judgment through substituting the value back into $$F''(a,b;c;x)$$ to find the local maximum according to $$.Get the graph and codes as below.

So we can get local critical points and make judgment as below:

In short, $$F(5,-10;1;x)$$ near $$x=0$$ has local maximum points which are$$x_1=0.2288 ,x_2=0.6834$$. All of these maximum points are at the points when $$x>0$$. When $$x<0$$, $$F(5,-10;1;x)$$ are monotonic increasing to $$\infty $$ as $$x\rightarrow -\infty $$.

Part 2
Prove equality

If we expand the right hand side of $$. Then we get

According to $$, noticing that it contains ther term $$b+k-1 $$and $$b $$begins with -10, so it would reach 0 at the 12th term and makes each terms afterwards cancelled to 0. As a resualt,the left hand side of $$could be written as

So We get $$proved.