Can a Periodic Function and Gamma Function Solve a Functional Equation?

[mathwonk]

TD: that approach in your book is the easiest of the three at elast if one propves the integrability of 1/x and the fundamental theorem of calculus.

now integrability is often not actually rpoved in most books, so to be honest that also is a gap in this so called easy approach.

what I sometime do in my course, instead of assuming integrability of continuous functions, which requires the concept of uniform continuity, is to prove integrability for monotone functions, which is much easier and covers the case of 1/x.

i.e. it is very easy to prove that the upper and lower Riemann sums for a monotone function on [a,b] converge to the same limit, as they differ by the product |f(b)-f(a)| times deltax, which obviously goes to zero as deltax does.

this is Newtons proof. then after knowing your monotone function is integrable, if it is also continuous it is very easy to prove the integral is differentiable, since the difference quotient [F(x+h)-F(x)]/h is bounded above by the area of the rectangle with base h and height f(x+h)-f(x) which approaches zero as h does by the definition of continuity.


If you look in stewart you will probably find that he assumes integrability of continuous functions or maybe proves it in an appendix that most courses skip.

 

[TD]

Well we didn't use one of the standard textbooks (such as Stewart etc), but one which was written by the professor himself (just for our course, non-commercial). We showed integrability of (piecewise) continuous functions, using upper and lower sums and, as you said, by relying on uniform continuity.

We also introduced (and showed it was possible to define) trigonometric functions in the same way: by defining the inverse functions as integrals, limiting them so they become bjiections and then define the inverse functions. It was interesting to see but as you remarked, not a very intuitive approach (although I don't think that was the point, just letting us see that it's possible and how you could rigorously introduce these things).

 

[mathwonk]

well the series approach is more advanced but it is nice, especially if motivated as i tried to do above. i overstated it when i said it should not be done that way, when what i should have said was it should be motivated first by convincing the reader that one is looking for a function which equals its own deriavtive. after that it is natural to use that definition.
as i said, it was done that way in my first freshman level calc course and i really liked it, as my appetite for rigor had never been met in high school trig courses.
Rudin of course cares nothing for motivation, and only for rigor and elegance, and although i respect his expertise, i do not enjoy his book for learning.
here is a nice application of that approach, which was something like problem 2 or 3 on one of our first homework assignments freshman year:
prove e defined as 1 + 1 + 1/2 + 1/3! + ... is irrational as follows:
assume e = n/m for some integers n,m>0 and get a contradiction as follows:
if it wereb true then em! would be an integer, but prove that for all m>0
em! is nevber and integer by direwct estimation.
i.e. multiply m! by the series for e, and look at the terms which are obviously integers, and estimate sum of the rest of the terms by comparing with a geometric series.
you wills ee that m!e equals an integer plus a term which is between 0 and 1, hence not an integer.

 

[mathwonk]

TD: I do like the approach via inverting integrals, as when generalized, it leads to the beautiful elliptic functions as inverses of the integral of 1/sqrt(1-x^4), and which are so important in algebraic geometry and number theory, e.g. in the proof of fermat's last theorem.
this approach aslso explains the trick of "separating variables" in solving d.e.'s.
i.e. by the inverse function theorem, if f' = 1/P(x), then g'(x) = P(g(x)), where g is the inverse of f.
this is the whole basis for the so called separable variables technique, but i have never seen it so simply explained in any book. i noticed it myself this fall while teaching integral calculus and discussing exactly the "inverse of integrals" ideas we have been discussing.


i.e. to solve dg/dx = P(g) or dy/dx = P(y), you separate variables, getting

dy/P(y) = dx and integrate both sides to get G(y) = x, and then

y = H(x) where H = inverse of G. i.e. if G'(x) = 1/P(x), then H' = P(H), where H is the inverse of G. this is of course just the inverse function theorem, but is usually presented as magic. (i hope i didn't screw this up too badly, but i ahve already correcetd several typos and mental errors. by the way notice that my posts are almost free of calculations and many others here above are almost entirely calculations. math is not really about calculation in my view. of course i am wrong, and merely compensating for my weak calculating ability.)

 

[benorin]

Rudin of course cares nothing for motivation, and only for rigor and elegance, and although i respect his expertise, i do not enjoy his book for learning.

Indeed! As a student presently learning from said text, I promptly cheered upon reading that. Thank you.

here is a nice application of that approach, which was something like problem 2 or 3 on one of our first homework assignments freshman year:
prove e defined as 1 + 1 + 1/2 + 1/3! + ... is irrational as follows:
assume e = n/m for some integers n,m>0 and get a contradiction as follows:
if it wereb true then em! would be an integer, but prove that for all m>0
em! is nevber and integer by direwct estimation.
i.e. multiply m! by the series for e, and look at the terms which are obviously integers, and estimate sum of the rest of the terms by comparing with a geometric series.
you wills ee that m!e equals an integer plus a term which is between 0 and 1, hence not an integer.

Nice.

 

[TD]

TD: I do like the approach via inverting integrals, as when generalized, it leads to the beautiful elliptic functions as inverses of the integral of 1/sqrt(1-x^4), and which are so important in algebraic geometry and number theory, e.g. in the proof of fermat's last theorem.
this approach aslso explains the trick of "separating variables" in solving d.e.'s.
i.e. by the inverse function theorem, if f' = P(x), then g'(x) = P(g(x)), where g is the inverse of f.
this is the whole basis for the so called separable variables technique, but i have never seen it so simply explained in any book. i noticed it myself this fall while teaching integral calculus and discussing exactly the "inverse of integrals" ideas we have been discussing.

We didn't really went any further than just defining the functions I described, as examples. We did see the inverse function theorem in the chapter before that, but we didn't prove it (it was said to be rather 'advanced' at that point). We used it though to prove the implicit function theorem (which was asked on the exam and I couldn't do it back then )

 

[mathwonk]

actually in one variable the inverse function theorem is quite easy, using only the intermediate value theorem (which tiself is deep of course, but usually assumed). you might try it.

in >1 varianble, the inverse function theorem is harder.

in my opinion learning from rudin is a bit masochistic, or sadistic, since the professor is calling the shots.

in general almost any book by simmons is readable, or apostol, or spivak, or courant, and if you are going to work that hard, why not go ahead and study dieudonne, and really learn the material deeply and learn much more too.

of course rudin also includes lebesgue integration theory.

what other real analysis books do people find readable, as alternatives to ("baby") rudin?

 

[benorin]

Apostol is cited alot, even as a secondary text (i.e. co-text) to Baby Rudin.

 

[mathwonk]

interesting since apostol is a freshman calc text and rudin is a senior junior analysis text. i agree, too, by the way and it went through ym mnid as i was writing my suggestions. apostol is outstanding, he gives a direct approach to constructing sin and cos as well which i have forgotten at the moment.

 

[benorin]

Personally, I enjoy Whittaker & Watson

 

[mathwonk]

isnt that mathematical physics?

 

[mathwonk]

nope, isee that is a 90 year old classic of "modern" analysis, written just when rigor was coming into vogue in britain, and contemporary with hardy. probably a wonderful source.

 

[mathwonk]

heres a copy from abebooks for 20 bucks.


4.
A Course of Modern Analysis. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions. American edition.
WHITTAKER, EDMUND TAYLOR, & WATSON, G. N.
Bookseller: J. Hawley Books
(Delanson, NY, U.S.A.) Price: US$*20.00
[Convert Currency] Shipping within U.S.A.:
US$*3.00
[Rates & Speeds]
Book Description: New York: The Macmillan Company, 1944 (reprint of the fourth edition, of 1927). 608 pages, index of authors, and general index., 1944. Hard cover. Good; cover worn, previous owner's name and bookplate on front endpaper and name stamp on rear endpaper. Bookseller Inventory # 1036

 

[benorin]

Bummer, I had just ordered one from amazon for ~$50.

 

[benorin]

As for pedagogy: this conversation has reminded me of a prior interest, namely the presentation of the gamma and beta functions. In particular, how exactly one defined the gamma function, whether it be as an integral or infinite product (Euler or Weierstrass) and with what motivation this is done. I was working on a paper which begins by defining the gamma function via direct continuation of the factorial via finite products->limit of a product->infinite products->integrals. I had been rather dilligent to ensure that my presentation was novel. Do you have a favored presentation of this topic?

 

[mathwonk]

i am not too familiar with the gamma function, but emil artin has a famous shoirt book on the topic. as i recall from 30 years ago he characterizes it as something loike the unique log convex extension of the afctorial??

and i think he sues te integral, but not sure.


the ww book i listed above at 20 bucks was used.

 

[benorin]

I have said book by Artin, and the characterization of the gamma function he uses is known as the Bhor-Mollerup Theorem.

 

[shmoe]

I'd say it depends on what properties of Gamma you are trying to emphasize. Euler's limit version of Gamma is probably the most natural to build up to if your goal was extending factorial.

Defining it as a function with poles at the non-negative integers (modulo some niceties) shows how it belongs in your stable of 'fundamental' meromorphic functions.

Defining it as an integral is probably the least motivated, except that this integral comes up in important places (e.g. zeta function) and it deserves to have a name of it's own. That it turns out to be an extension of factorial is kind of an unasked for byproduct with this view (same with the product over the poles). This has the bonus of being simple enough for a first year calculus student to understand, so it's natural to be the first one a student sees and I don't thnk this is a bad thing.

The Bohr-Mollerup characterization is one of those after the fact things that's hard to justify as a starting point, and you end up using one of the usual definitions to prove this unique function actually exists. How do you justify this log-convex condition as being a 'natural' one apart from the fact that it 'works'?

 

[benorin]

Well, if interpolation of the factorial is your only goal, then the functional equation f(n+1)=nf(n),\mbox{ for }n=1,2,3,\ldots being satisfied would suffice; yet a solution to such is not unique, both the gamma function and the Barnes G-function are solutions to the above functional equation.

 

[benorin]

Here's one right on topic

A problem: From the following

i. \lim_{q\rightarrow\infty}\prod_{n=p}^{pq} \left(1+\frac{x}{n}\right)=p^{x},

and

ii.\lim_{n\rightarrow\infty}\left(1-\frac{t}{n}\right)^{n}=e^{-t},

reason that

\int_{t=0}^{\infty}e^{-t}t^{x}dt=\lim_{n\rightarrow\infty}\frac{1\cdot 2\cdots n}{(1+x)(2+x)\cdots (n+x)}n^{x},

where x is complex and not a negative integer.

I quoted this exercise from Introduction to the Theory of Analytic Functions by Harkness & Morley that I happed to have a single page of printed (pg. 208), (i) is the from a separate exercise listed immediately prior to the exercise at hand in which (ii) is a given.

 

[shmoe]

Well, if interpolation of the factorial is your only goal, then the functional equation f(n+1)=nf(n),\mbox{ for }n=1,2,3,\ldots being satisfied would suffice; yet a solution to such is not unique, both the gamma function and the Barnes G-function are solutions to the above functional equation.

Doesn't the Barnes function grow much faster and satisfy G(n+1)=Gamma(n)*G(n)? How can this possibly interpolate factorial? Barnes is the one with zeros of order of order |n| at negative integers n isn't it?

Of course there's lots of ways to extend factorial, you can always connect the dots. My point was if you were introducing gamma with this goal, then the limit definition would involve less "where on Earth did that come from". To me it requires the least motivation via hindsight.

 

[benorin]

My bad, bad memory that is: If u(x) is any function such that \forall x,u(x+1)=u(x) (i.e. u is has a period of unity,) then u(x)\Gamma (x) is a solution to said functional equation.