Wikipedia:Reference desk/Archives/Mathematics/2008 July 1

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July 1

Analyticity on R

We already know that all of the derivatives of $e^{x}$ at $x=0$ are equal to 1. Find another analytic function on $\mathbb {R}$, call it $f$, that is not equal to $e^{x}$ at any $x$ other than $x=0$, but for which the $n$th derivative of $f$ at 0, $f\,^{(n)}(0)$ is equal to 1 for all $n\geq 1$.

Now I thought that an analytic function has a power series expansion, the expansion is unique, and the coefficients are determined by the derivatives. In addition, the function would be equal to its expansion if the remainder term goes to zero. Now, if we find the answer to the above question, wouldn't that imply that the same power series represents two different functions? Because the coefficients of our f(x) and $e^{x}$ will be the same (if we expand around x=0). Is there something subtle that I am missing here or is the answer obvious but I am having one of my (too many) dumb moments? The question is verbatim from an analysis book. What would such a function look like? Thanks--76.79.202.34 (talk) 00:39, 1 July 2008 (UTC)

If you're allowed to have f(0) ≠ 1, then you can take f(x) = e^x + 1. If not, then I agree it's impossible. On the other hand, if what you're looking for is not an analytic function, but an infinitely differentiable one, then it is possible. I think e^x + e^-1/x2, suitably extended at 0, does the trick. The thing is, analytic and infinitely differentiable are synonymous for complex functions, but not real ones. 136.152.224.6 (talk) 03:52, 1 July 2008 (UTC)

Indeed. We have an article on functions that are infinitely differentiable but not analytic. In particular, there are non-analytic functions whose derivatives of all orders are equal to 0 to x=0 (but which are not the zero function); if the question allowed non-analytic functions, you could add one of these to e^x. But since the question requires an analytic function, I would say it is impossible - but the point of the exercise might be to get you to think about why it is impossible ... Gandalf61 (talk) 10:29, 1 July 2008 (UTC)

Interesting! Now just for curiosity, are there any other examples of real functions which are infinitely differentiable but are not equal to its power series expansion? I only knew of one example, the famous $e^{-{\frac {1}{x^{2}}}}$ and then the piecewise function
$f(x)={\begin{cases}\exp(-1/x)&{\text{if }}x>0,\\0&{\text{if }}x\leq 0,\end{cases}}$
described in the article.--76.79.202.34 (talk) 23:41, 1 July 2008 (UTC)

You can create lots of examples by modifying and combining the above. $\sum _{k=1}^{7}e^{-{x^{-2k}}}.$ Bo Jacoby (talk) 11:46, 2 July 2008 (UTC).

I think that you can construct a whole family of smooth but non-analytic functions by generalising these examples as follows:

$f(x)={\begin{cases}\exp(-1/x^{n})&{\text{if }}x>0,\\0&{\text{if }}x=0,\\\exp((-1)^{m+1}/x^{m})&{\text{if }}x<0,\end{cases}}$

where n and m are positive integers; and you can also replace either "side" of any of these function by f(x) = 0. These functions should all have derivatives of all orders that are equal to 0 at x=0 (verifying this is left as an exercise to the reader !).

Smooth but non-analytic functions are in some senses more "common" than analytic functions - our article on smooth functions says "Although it might seem that such functions are the exception rather than the rule, it turns out that the analytic functions are scattered very thinly among the smooth ones; more rigorously, the analytic functions form a meagre subset of the smooth functions". But possibly, like transcendental numbers, it may be difficult to find many "natural" and easily defined examples. Gandalf61 (talk) 09:09, 2 July 2008 (UTC)

Awesome, thanks everyone who answered.--A Real Kaiser (talk) 23:58, 2 July 2008 (UTC)

BTW, you can use convolution with a bump function (that exp(-1/x^2) guy) to smooth any function. It barely changes it, but makes it infinitely differentiable. This is often seen in a PDE course where you smooth a continuous "harmonic" function (or a harmonic measure, or a harmonic distribution, etc.) and notice that due to the mean value property it is equal to its regularization (almost everywhere), so was itself infinitely differentiable. At any rate, if you had a function you wanted to be C^oo, try regularizing it. JackSchmidt (talk) 21:15, 3 July 2008 (UTC)

FALSE POSITION

METHOD OF FALSE POSITION FOR FINDING A REAL ROOT OF A NONLINEAR EQUATION? —Preceding unsigned comment added by Wikifrommld (talk • contribs) 03:15, 1 July 2008 (UTC)

• I will answer your "question" with the link to the false position method article. If you have an actual question, then feel free to ask it... --Kinu ^t/_c 04:40, 1 July 2008 (UTC)

ALGORITHM

A CONCISE ALGORITHM OF THE FALSE POSITION METHOD —Preceding unsigned comment added by Wikifrommld (talk • contribs) 03:17, 1 July 2008 (UTC)

Already given above. Look at the article. Or look it up on Google. 76.217.110.126 (talk) 05:40, 1 July 2008 (UTC)

Please have your keyboard repaired — the CapsLock key needs replacing. --CiaPan (talk) 06:15, 1 July 2008 (UTC)

I think caps-lock and bold looks quite acceptable - smart and concise.87.102.86.73 (talk) 09:49, 1 July 2008 (UTC)

Or maybe one of the shift-keys is stuck? 195.35.160.133 (talk) 14:53, 1 July 2008 (UTC) Martin.

Can't answer a problem that I made up!

I made up the problem below:

$\lim _{x\to 90}{\dfrac {\sin(90-x)}{\cos x}}$

Is it possible to solve it? Alexius08 is welcome to talk about his contributions. 06:17, 1 July 2008 (UTC)

Assuming you're using degrees instead of radians, the limit is 1 because sin(90° – x) = cos(x) for all x. If you're using radians, the limit is just 0 by direct substitution. 76.217.110.126 (talk) 06:28, 1 July 2008 (UTC)

But (applying the limit) ${\dfrac {\sin(90-x)}{\cos x}}$ results into:

$={\dfrac {\sin(90-90)}{\cos 90}}$

$={\dfrac {\sin 0}{\cos 90}}$

$={\dfrac {0}{0}}$

Therefore, it is indeterminate. Alexius08 is welcome to talk about his contributions. 06:35, 1 July 2008 (UTC)

Wrong. Applying the limit yields
    $\lim _{x\to 90}{\dfrac {\sin(90-x)}{\cos x}}=\lim _{x\to 90}{\dfrac {\cos x}{\cos x}}=\lim _{x\to 90}1=1,$
which surely is not 'indeterminate'. --CiaPan (talk) 07:01, 1 July 2008 (UTC)

An interesting observation is that, had the unit been in radians, then the limit would undisputedly be zero, even by Alexius08's reasoning, this is because while sin(90-90) = sin0 = 0, cos90, when 90 means 90 rad, is most definitely not zero. 74.12.198.105 (talk) 21:32, 1 July 2008 (UTC)

I believe you could treat either form as an indeterminate and use L’hopitals rule. It provides a nice infinite loop, and sometime you have to realize there is some cancelation. But I don’t think you need to do that from the get go.

$\lim _{x\to 90}{\frac {\sin(90-x)}{\cos x}}=\lim _{x\to 90}{\frac {{\tfrac {d}{dx}}(\sin(90-x))}{{\tfrac {d}{dx}}(\cos x)}}=\lim _{x\to 90}{\frac {-\cos(90-x)}{-\sin x}}$

GromXXVII (talk) 10:53, 1 July 2008 (UTC)

For a similar but simpler case, think of taking the limit as x approaches 0 of x/x. Now the function x/x does not have a value at x=0, but it has a perfectly good two sided limit as x approaches 0, which is 1. Gandalf61 (talk) 10:06, 1 July 2008 (UTC)

Just a clarification: I used degrees as the units of measurement on this case. Alexius08 is welcome to talk about his contributions. 09:14, 3 July 2008 (UTC)

Ah, I see, GromXXVII. Alexius08 is welcome to talk about his contributions. 09:22, 3 July 2008 (UTC)

Fractional Calculus

After reading the fractional calculus article I wanted to understand fractional derivatives better, so tried to find one of a more complex function. I used one which would require product rule, partly out of curiosity to see how it would turn out. The only method shown in the article to obtain fractional derivatives is effectively pattern spotting, so thats how I tackled, however I have come to a problem

I started with the function

$f(x)=x^{k}e^{nx}\,$

so

${\frac {d}{dx}}f(x)=kx^{k-1}e^{nx}+x^{k}ne^{nx}\,$

${\frac {d^{2}}{dx^{2}}}f(x)=k(k-1)x^{k-2}e^{nx}+2kx^{k-1}ne^{nx}+x^{k}n^{2}e^{nx}\,$

soon a pattern emerges

${\frac {d^{a}}{dx^{a}}}f(x)=\sum _{r=0}^{a}{\binom {a}{r}}{\Big (}{\frac {k!}{(k-r)!}}x^{k-r}{\Big )}{\Big (}n^{a-r}e^{nx}{\Big )}$

the first part is the binomial coefficient, then the $x^{k}$ term differentiated $r$ times then the $e^{nx}$ term differentiated $a-r$ times. You can express binomial coefficients and factorials in terms of the gamma function to allow for non integer terms, however as far as I can see, this does not have the power of being able to give fractional derivatives due to the fact it is given as a summation series. Is there another way this could be expressed, possibly as an integral(?), that would allow for fractional derivatives (i.e. where a non integer value of $a$ is used) to be calculated? Thanks. Philc 0780 11:35, 1 July 2008 (UTC)

I've no idea - but - you seem to need 'fractional polynomials'..

..now I know that hypergeometric series and the like (ie extensions of such as a_n=k₁a_n-1+k₂a_n-2 ) produce finite polynomials for certain values (and infinite polynomials for others), perhaps if you express the coefficients of the polynomials in 'hypergeometric series' form - you could get the series for non-integers.. (Just a guess - don't blame me when it all goes horribly wrong).87.102.86.73 (talk) 13:03, 1 July 2008 (UTC) I don't think that would work, hopefully it might give you another idea.87.102.86.73 (talk) 14:29, 1 July 2008 (UTC)

(Having another try - hopefully someone else will turn up..) —Preceding unsigned comment added by 87.102.86.73 (talk) 15:09, 1 July 2008 (UTC)

Since you have the binomial coefficient included in the terms for x^whatever it's worth noting that the numerator takes the form

Product(from a to a-r) = a(a-1)(a-2)...(a-r+1)

Now if r=a+1 the numerator becomes zero... making all further terms zero as well.

So if you could somehow refactor the binomial into a form where the numerator is zero beyond r=a then you can replace your sum from (which is from 0 to a) with a sum from zero to infinity..

This might help simplify things, since then you don't have the problem of trying to take a sum to a non-integer value??87.102.86.73 (talk) 15:08, 1 July 2008 (UTC)

ie I'm suggesting that you take ${\binom {a}{r}}$ to be zero when r>a and sum from 0 to infinity...87.102.86.73 (talk) 15:11, 1 July 2008 (UTC)

if you do the above you should get infinite polynomials in x when a is non-integer.. no idea if those would be the right answer though. (don't even know if you'll get a convergent series.)87.102.86.73 (talk) 15:18, 1 July 2008 (UTC)

Writing this way

$y=e^{nx}x^{k}\,$

${\frac {dy}{dx}}=e^{nx}x^{k}n(1+{\frac {k}{nx}})\,$

${\frac {d^{2}y}{dx^{2}}}=e^{nx}x^{k}n^{2}(1+2{\frac {k}{nx}}+{\frac {k(k-1)}{(nx)^{2}}})\,$

${\frac {d^{a}y}{dx^{a}}}=e^{nx}x^{k}n^{a}\sum _{r=0}^{\infty }{\binom {a}{r}}{\binom {k}{r}}{\frac {r!}{(nx)^{r}}}$

If k is a nonnegative integer, the sum is finite even when a is fractional. So this function

$f(a,n,k,x)=e^{nx}x^{k}n^{a}\sum _{r=0}^{\infty }{\binom {a}{r}}{\binom {k}{r}}{\frac {r!}{(nx)^{r}}}$

is defined for all complex values of a and x, and positive values of n, and nonnegative integer values of k, and satisfies

$f(a,n,k,x)={\frac {d^{a}(e^{nx}x^{k})}{dx^{a}}}$

when a happens to be a nonnegative integer, where the right hand side is defined. Bo Jacoby (talk) 23:15, 1 July 2008 (UTC).

Thank you for your help, but the fact that $a$ has to be a natural number is the problem I'm concerned with, you say it is only defined for integer values of $a$ and $k$, is that only due to the binomial coefficients, because by substitution of the gamma function we can use any real or complex numbers in calculation of those coefficients, and are not restricted to natural numbers. Would that allow it to be defined for all real $a$, as that is what fractional calculus is concerned with. Philc 0780 10:51, 2 July 2008 (UTC)

Yes. When $r$ is a nonnegative integer, then ${\binom {a}{r}}$ is defined for all real (and all complex) values of $a.$ If $k\in \{0,1,\cdots ,r-1\},$ then ${\binom {k}{r}}=0.$ So the infinite series has only a finite number of nonzero terms, and the formula should be written

$f(a,n,k,x)=e^{nx}x^{k}n^{a}\sum _{r=0}^{k}{\binom {a}{r}}{\binom {k}{r}}{\frac {r!}{(nx)^{r}}}.$

So the function $f$ is defined for all real $a$, and for integer values of $a$ it equals the derivative no $a$, so you may use $f$ for defining a fractional derivative. Bo Jacoby (talk) 12:32, 2 July 2008 (UTC).

edit conflict - comment probably now unneeded. 'Bo Jacoby' said If k is a nonnegative integer, the sum is finite even when a is fractional which is true, all that happens when k AND a are not integers is that the series in infinite - I see no problem with that..

Effectively what both of us have said is that; assuming the equation you provided is correct (which I am sure is), for the whole derivatives you can actually ignore the sum from r=0 to a and replace with r=0 to infinty since the sum is identical for integer values of a,k etc..

So then you may as well assume that this fuction is good for the fractional derivatives.. if you doubt this I could only suggest creating

d^r/dx^r(d^q-r/dx^q-r(f(x) )

expanding the terms, collecting powers of x and hopefully being able to show that this equals the q^th derivative for all r.. (Maybe there is an easier way to confirm the equation is correct)87.102.86.73 (talk) 12:47, 2 July 2008 (UTC)

Statistics - use of 'linear by linear'

I am a student checking to see if there is a relationship between the presence of DNA methylation for a particular gene and tumour grade. In SPSS, methylation vs non-methylation are encoded as 1 and 0 respectively (numerical, nominal), while the tumour grades are 1, 2, 3 and 4 (numerical, ordinal). SPSS is giving me a result "linear-by-linear" analysis but if I understand correctly, I can't use this because methylation/non-methylation are not numerical factors. Is that right? ----Seans Potato Business 14:09, 1 July 2008 (UTC)

According to this website: "Mantel-Haenzel chi-square is not appropriate for nominal variables.". I think, given the amount charged for the SPSS program, it should be smart enough to not calculate and present this statistic since I have specified that the methylation/non-methylation field is nominal... ----Seans Potato Business 15:16, 1 July 2008 (UTC)

If the methylation scale is 0 or 1, is there any reason for not taking this as ordinal? I'd be inclined to take the other variable as interval, but maybe I don't know enough about how the tumours are categorised.…81.154.108.9 (talk) 22:39, 1 July 2008 (UTC)

Assuming you are contructing a linear regression model, if the methylation is binary ordinary least squares is not appropriate. You need to estimate the relationship via a logit or probit procedure. Wikiant (talk) 12:33, 2 July 2008 (UTC)