User:Oh Isaac/Hypergeomtric Function

This report is a summary of the discussion between Professor Loc Vu-Quoc and Isaac Lin about hypergeometric functions.

 there are many misprints in this report because of careless writing. reread every word carefully to correct all misprints. Egm3520.s13 (discuss • contribs) 11:35, 25 February 2013 (UTC)

Given
Hypergeomtric function could be written as a series:

in which:

$$(q)_n = \left\{ \begin{array}{ll} 1                    & \mbox{if } n = 0 \\ q(q+1) \cdots (q+n-1) & \mbox{if } n > 0 \end{array} \right .$$

provided that c does not equal to 0, −1, −2, ... . Here (q)n is called the Pochhammer symbol.Notice that the series terminates if either a or b is a nonpositive integer. As to complex arguments z with |z| ≥ 1, it can be analytically continued along any path in the complex plane that avoids the branch points 0 and 1.

 you need to provide links to various concepts, e.g., Pochhammer symbol, analytical continuation, etc. Egm3520.s13 (discuss • contribs) 12:01, 25 February 2013 (UTC)

Oh Isaac (discuss • contribs) 05:49, 27 February 2013 (UTC)
 * Revised

Find
In question R5.11 of in the course of EGM 6321 Principles of Engineering Analysis 1, programs made by Matlab code show that the formula $$\,_2F_1(\frac{1}{3},1;\frac{4}{3},x)$$ has imaginary parts, which is also shown in the website of WolframAlpha as below.

 your figure does not have a license; you need to include a license to avoid it being deleted in the future. Egm3520.s13 (discuss • contribs) 22:36, 7 March 2013 (UTC)

Oh Isaac (discuss • contribs) 01:03, 8 March 2013 (UTC)
 * Revised

This leads to the central question: How could the hypergeometric function, which can be expressed as the sum of a real number series, has complex part?

Background Knowledge
 there are many misprints because of careless writing. reread every word carefully to correct all misprints. Egm3520.s13 (discuss • contribs) 11:35, 25 February 2013 (UTC)

The Hypergeometric function$$F(a,b;c;x)$$satisfies the hypergeometric equation:

In the equation the argument, x, and three paramenters, a,b, and c, are not restricted to being real only. This equation has three regular singular points at $$z=0,1$$and $$\infty$$, unless certain relations are applied to three parameters.

Near the point $$z=0$$ the hupergeometric equation has two independent solutions, one is the series expansision of $$, but a more formal form of the items in it is:

where $$\Gamma$$is the Gamma function. Gamma function defined as$$\Gamma(z) = \int_0^\infty t^{z-1} e^{-t}\,{\rm d}t$$to all complex numbers except the non-positive integers (where the function has simple poles).

 do you understand the gamma function (many readers don't). provide a brief description and refs, including web links such as wikipedia articles. also, you need to give detailed refs to all equations, which you clearly did not derive; you copied from other refs without giving credit; that's plagiarism. Egm3520.s13 (discuss • contribs) 11:35, 25 February 2013 (UTC)

Oh Isaac (discuss • contribs) 09:02, 27 February 2013 (UTC)
 * Revised

no, the definition of the gamma function is more general than that. what you wrote was the definition for integers only; the gamma function was also defined for any real number. it is a generalization of the factorial operation for integers to real numbers. Egm3520.s13 (discuss • contribs) 22:39, 7 March 2013 (UTC)

Oh Isaac (discuss • contribs) 00:01, 8 March 2013 (UTC)
 * Revised

Another solution of the hypergeomtric equation near 0 is

 you clearly did not derive this equation; you copied it from other sources (papers, books, etc.), which you need to give detailed ref (author's name, article title, journal bibliographic details, including page number and equation number for this equation in that ref). do the same for ALL equations in this report. Egm3520.s13 (discuss • contribs) 12:01, 25 February 2013 (UTC)

Oh Isaac (discuss • contribs) 09:02, 27 February 2013 (UTC)
 * Revised

your reference to the NIST web site is too broad; you need to be more specific in your reference (e.g., which web page, which equation number, etc.) to help the reader to quickly find the exact location where this equation was written. Egm3520.s13 (discuss • contribs) 22:49, 7 March 2013 (UTC)

Oh Isaac (discuss • contribs) 01:25, 8 March 2013 (UTC)
 * Revised

The series $$converges for |x|<1, and could be expressed as a uniformly converging integral ,

where$$Re[c]>Re[b]>0, |x-1|>0, \mbox{arg}(1-x)<\pi.$$

 you need to give the precise reference where Eq.(5) appeared. Egm3520.s13 (discuss • contribs) 22:49, 7 March 2013 (UTC)

Oh Isaac (discuss • contribs) 01:54, 8 March 2013 (UTC)
 * Revised

 you need to provide the full bibliographic detail for the reference by Hsu (1993); did you actually read this paper ? or did you just get this ref from the other paper by Forrey (1997) that you did provide full bibliographic detail ? Egm3520.s13 (discuss • contribs) 11:35, 25 February 2013 (UTC)

Oh Isaac (discuss • contribs) 09:02, 27 February 2013 (UTC)
 * I have the paper of Hsu (1993) and all other papers I used in this article from the database of UF, I wrote the full bibliographic detail at the end .Should I upload it to this website?

no, see my e-mail to you on this issue.

correct

"Accoring to YUPAI P. HSU"

to

"According to Hsu (1993)"

Egm3520.s13 (discuss • contribs) 22:49, 7 March 2013 (UTC) Oh Isaac (discuss • contribs) 01:28, 8 March 2013 (UTC)
 * Revised

According to HSU(1993), the integral in $$is analytic in x for every t. Accordingly, the hypergeometric function, $$F(a,b;c;x)$$is analytic in the whole x-plane cut along $$[0,\infty]$$, not just within the original |x|<1. Therefore it can be extended to the whole x-plane. Regarded as a function of the three complex paramenters, the hypergeometric funciton is an entire function of both a and b, and a meromorphic function of c, with only simple poles at $$c=0,-1,-2,\cdots$$, which is shown in $$caused by the singularities of parameters in gamma function $$\Gamma(c)$$.

 don't just copy from other papers such as Forrey (1997); if you don't understand what "meromorphic" is (and many readers don't), you need to explain. don't pretend that you understood everything that you read in Forrey (1997) or in other sources. provide a brief description and refs, including web links such as wikipedia articles. Egm3520.s13 (discuss • contribs) 11:35, 25 February 2013 (UTC)

Oh Isaac (discuss • contribs)
 * Well it is true I don't really understand "Meromorphic", it is mentioned in Hsu's paper. I copied it here to avoid any unknown mistakes. I have linked it to its wiki page.

The hypergeometric function can be analytically extended to other regions from within the unit disk |x|<1 by substituting the original argument to other forms. For example, by substituting t to 1-s, $$can be transformed to

 not "transfer"; change to "transform". Egm3520.s13 (discuss • contribs) 12:01, 25 February 2013 (UTC)

Oh Isaac (discuss • contribs) 09:02, 27 February 2013 (UTC)
 * Revised

In order to keep convergence, we need $$|\frac{z}{z-1}|<1$$, for complex number z

So the available z extends to any $$Re[z]<\frac{1}{2}$$with $$|\frac{z}{z-1}|<1$$. On the right hand side of $$a convergent series is obtained which can be used to calculate the function value. There are twenty four valid transformations like $$obtained by Kummer, called Kummer's Solution.

To do this, the real axis was divided into six intervals and in each interval the argument w is strictly limited in the convergence range so that the series in powers of w is kept converged rapidly, which are shown in the table below. Details of the whole transformations could refer to Rorbert C. Forrey.

 it is not clear why to any $$Re[z]<\frac{1}{2}$$, there is $$|\frac{z}{z-1}|<1$$. you need to explain in detail. Egm3520.s13 (discuss • contribs) 12:01, 25 February 2013 (UTC)

 you clearly scan this table from some source but did not refer to. also you did not provide a license, so this figure will be deleted in the future; that's irresponsible since this report will be unreadable. it is better to retype the table and provide detailed ref (page number in which article). Egm3520.s13 (discuss • contribs) 12:01, 25 February 2013 (UTC)

Oh Isaac (discuss • contribs) 09:02, 27 February 2013 (UTC)
 * Revised. We didn't provide any licence for the pictures in our class reports,will they be deleted in the future?

Explanation of the Problem
In the problem referred to previously, $$\,_2F_1(\frac{1}{3},1;\frac{4}{3},x)$$ comes to be a complex value in the domain of $$x>2$$. In this case, when calculating the value of it, the substitution of argument is from x to $$\frac{1}{x}$$, and the transformation formula is

with constraints that

 change "restraints" to "constraints". Egm3520.s13 (discuss • contribs) 12:01, 25 February 2013 (UTC)

Oh Isaac (discuss • contribs) 09:14, 27 February 2013 (UTC)
 * Revised

so we get a convergent series to calculate the value when |x|>2.According to $$i=\sqrt{-1}$$, once a or b is a proper fraction and x is positive,$$will generate an imaginary part from $$(-x)^{a}$$or $$(-x)^{b}$$. This can explain why complex parts appear in the problem.

 what you wrote below is not clear:

$$will generate imaginary part, once a or b is proper fraction and x is positive, so are they in the problem, which makes a complex value from $$(-x)^{\frac{1}{n}}$$(x and n are positive integers), according to $$i=\sqrt{-1}$$. So it can explain why complex parts appear.

you need to improve the written English in this report as well as the explanation.

also, there is something wrong with your use of "EquationRef"; there is no blank space possible after "EquationRef" and the next word.

Egm3520.s13 (discuss • contribs) 12:01, 25 February 2013 (UTC)

Oh Isaac (discuss • contribs) 09:20, 27 February 2013 (UTC)
 * Revised

Conclusion
Therefore we know from above that the hypergeometric series $$is an expression to represent the hypergeometric function when argument x is limited in |x|<1, but is not the only one to describe it. $$and its transformations are also used to describe the hypergeometric function in different domains and the transformations may generate complex value. All of the above are solutions of the hypergeometric equation.

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Trial Explanation 1
by Loc Vu-Quoc

i thought that there could be a series of real numbers that converges to an imaginary number since $$i=\sqrt{-1}$$. so if you consider the power series of, say, $$(1+x)^{-1/2}$$, and let $$x\rightarrow-2$$, then you would get the imaginary unit i. you may want to think along this line.

of course, you had an example of an infinite series of real numbers converging to complex numbers: Review the exponential of a matrix lecture in which i gave an example of a real matrix whose exponent contained complex numbers (oscillations).

re the converged result of a series as a complex number, think of the exponential of a matrix for the case of oscillations that i mentioned to you and that i had in my lecture notes. if you truncate the series (no matter how high the number of terms you took, 400, 500, 1 million, 1 trillion, etc.), you would always get a real matrix ! the converged series, i.e., with infinitely many terms, is a matrix with complex coefficients.

Problem
by Lin

I understand the the exponential of a matrix oscillates because it multiplies by i by itself, but in this case it seems to involved more complex matrix computations and we should have a way to find the imaginary part in it like we can use Euler formula to separate sin and cos into the form contains i. But in the $$I don't find any component of i. I don't know whether I understand your meaning.

$$expm\left[\begin{vmatrix} 0&-1 \\ 1&0 \end{vmatrix}t\right]=\begin{vmatrix} \cos{x} & -\sin{x}\\ \sin{x} & \cos{x} \end{vmatrix}=\begin{vmatrix} \frac{e^{-ti}+e^{ti}}{2}& \frac{-e^{-ti}+e^{ti}}{2}i\\ \frac{e^{-ti}-e^{ti}}{2}i&\frac{e^{-ti}+e^{ti}}{2} \end{vmatrix}$$

the value of this matrix also contains imaginary part when t change:

Trial Explanation 2
by Lin

According to Rorbert C. Forrey, the hypergeometric fuction$$\,_2F_1$$ is defined by series $$, which converges for$$|x|<1 $$as can be seen by the ratio test. For z outside the circle of convergence, it is necessary to transform z into a new form of hypergeomtric function with argument w that $$|x|<1 $$.

To do this, the real axis was divided in to six intervals and in each interval the argument w is strictly limited in the convergence range so that the series in powers of w is kept converged rapidly.

For example:

When $$-\infty<x<-1$$, the transformation of argument is $$w=\frac{1}{1-x}$$, and the transformation of function is:

In this way, the range is limited and divergence is almost avoid. (They also use finite difference and similar function to avoid infinite and divergence caused by Gamma function. The Gamma function and other transformation function may generate imaginary part). So the value of hypergeomtric function is not calculated only by the sum of a series of real numbers but get involved with complex functions, which made the result complex when software uses this method to calculate the value.

Trial Explanation 3
by Lin

Maybe it is just a extension tips automatically caused by the computer or website. Consider a series:

And we know:

The WolframAlpha shows that this series has a imaginary part. But as we know, $$diverge in the range $$|x|>1$$,so it has no value. But WolframAlpha show a complex part of it which in fact should be the complex graph of $$, not the original series.

Problem
by Lin I think the value of andare equal just in a particular range, they themselves are not the same thing. But when I chose "real-value plot" in WolframAlpha the imaginary part disappeared.I don't know what is the difference.

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