Who was it who figured out that integral 1/(x^2 + 1) = arctan x ? And how long did it take them before they published, and did their colleagues think they were nutso at first? It's been over 30 years since I learned this stuff, and having trigonometric functions appear from ratios of polynomials still astonishes me, every time. -- www.wescottdesign.com
Math History
Started by ●October 9, 2014
Reply by ●October 9, 20142014-10-09
On 2014-10-09 16:08:17 +0000, Tim Wescott said:> Who was it who figured out that > > integral 1/(x^2 + 1) = arctan x > > ? > > And how long did it take them before they published, and did their > colleagues think they were nutso at first? > > It's been over 30 years since I learned this stuff, and having > trigonometric functions appear from ratios of polynomials still > astonishes me, every time.I expect the way to go is to demonstrate what the derivative of arctan is. That then yields the integral. To get the derivative of arctan you combine the derivative of the inverse function, via the implicit function theorem, with the derivative of tan. So you just decompose the problem into smaller pieces. As always, the trick is finding the right pieces.
Reply by ●October 9, 20142014-10-09
>Who was it who figured out that > >integral 1/(x^2 + 1) = arctan x > >? > >And how long did it take them before they published, and did their >colleagues think they were nutso at first? > >It's been over 30 years since I learned this stuff, and having >trigonometric functions appear from ratios of polynomials still >astonishes me, every time. > >-- >www.wescottdesign.com >For any of you guys interested in math history, you should check out the book "An Imaginary Tale: The Story of [the square root of minus 1]". That's pretty cool stuff. -Doug _____________________________ Posted through www.DSPRelated.com
Reply by ●October 9, 20142014-10-09
Tim Wescott <tim@seemywebsite.com> writes:> Who was it who figured out that > > integral 1/(x^2 + 1) = arctan x > > ? > > And how long did it take them before they published, and did their > colleagues think they were nutso at first? > > It's been over 30 years since I learned this stuff, and having > trigonometric functions appear from ratios of polynomials still > astonishes me, every time.This has led me to some interesting web searches. It seems that the elements of calculus were known in geometric terms prior to their expression in algebraic terms by Newton and Leibniz. Anyway, James Gregory published a version of the fundamental theorem of calculus in 1668(1)(2). He developed the series expansion for arctan in 1671 (or earlier)(2). His expansion was based on the the derivative of the arctangent which discovered previously by "others"(3). So I would say this result was known before calculus was put on a modern foundation by Newton and Leibniz(4). (1)http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus (2)http://en.wikipedia.org/wiki/James_Gregory_%28mathematician%29 (3)http://www.math.wpi.edu/IQP/BVCalcHist/calc3.html (4)http://en.wikipedia.org/wiki/History_of_calculus Scott -- Scott Hemphill hemphill@alumni.caltech.edu "This isn't flying. This is falling, with style." -- Buzz Lightyear
Reply by ●October 9, 20142014-10-09
Scott Hemphill <hemphill@hemphills.net> writes:> Tim Wescott <tim@seemywebsite.com> writes: > >> Who was it who figured out that >> >> integral 1/(x^2 + 1) = arctan x >> >> ? >> >> And how long did it take them before they published, and did their >> colleagues think they were nutso at first? >> >> It's been over 30 years since I learned this stuff, and having >> trigonometric functions appear from ratios of polynomials still >> astonishes me, every time. > > This has led me to some interesting web searches. It seems that the > elements of calculus were known in geometric terms prior to their > expression in algebraic terms by Newton and Leibniz. > > Anyway, James Gregory published a version of the fundamental theorem of > calculus in 1668(1)(2). He developed the series expansion for arctan in > 1671 (or earlier)(2). His expansion was based on the the derivative of > the arctangent which discovered previously by "others"(3). So I would > say this result was known before calculus was put on a modern foundation > by Newton and Leibniz(4). > > (1)http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus > (2)http://en.wikipedia.org/wiki/James_Gregory_%28mathematician%29 > (3)http://www.math.wpi.edu/IQP/BVCalcHist/calc3.html > (4)http://en.wikipedia.org/wiki/History_of_calculusOh, I forgot to mention: this result might have been known centuries earlier by Madhava of Sangamagrama (c. 1350 – c. 1425)(5) (5)http://en.wikipedia.org/wiki/Madhava_series Scott -- Scott Hemphill hemphill@alumni.caltech.edu "This isn't flying. This is falling, with style." -- Buzz Lightyear
Reply by ●October 9, 20142014-10-09
On Thu, 09 Oct 2014 11:08:17 -0500, Tim Wescott <tim@seemywebsite.com> wrote:>Who was it who figured out that > >integral 1/(x^2 + 1) = arctan x > >? > >And how long did it take them before they published, and did their >colleagues think they were nutso at first? > >It's been over 30 years since I learned this stuff, and having >trigonometric functions appear from ratios of polynomials still >astonishes me, every time. > >-- >www.wescottdesign.comSin and cos can both be represented by Taylor series expansions, and tan is the ratio of sin and cos. So it's not too surprising from that standpoint. Eric Jacobsen Anchor Hill Communications http://www.anchorhill.com
Reply by ●October 9, 20142014-10-09
Tim Wescott <tim@seemywebsite.com> wrote:> Who was it who figured out that> integral 1/(x^2 + 1) = arctan x> And how long did it take them before they published, and did their > colleagues think they were nutso at first?> It's been over 30 years since I learned this stuff, and having > trigonometric functions appear from ratios of polynomials still > astonishes me, every time.As well as I know it, figuring out that integral dx/x was a pretty important part of the development of calculus. I am not so sure about this one. -- glen
Reply by ●October 10, 20142014-10-10
Tim Wescott <tim@seemywebsite.com> wrote:> Who was it who figured out that> integral 1/(x^2 + 1) = arctan xThe book "The History of the Calculus and its Conceptual Development" gives a pretty good description of the way things actually were done. Most books are arranged to make it easier to learn, which is very different from the historical development. I don't see this specific question, though. -- glen
Reply by ●October 10, 20142014-10-10
On Thursday, October 9, 2014 12:08:17 PM UTC-4, Tim Wescott wrote:> Who was it who figured out that > > integral 1/(x^2 + 1) = arctan x > > ? >probably the same person that figured out that (d/dx) arctan(x) = 1/(x^2 + 1)> And how long did it take them before they published, and did their > colleagues think they were nutso at first? > > It's been over 30 years since I learned this stuff, and having > trigonometric functions appear from ratios of polynomials still > astonishes me, every time.i dunno. i was sorta astonished first time i saw Euler's formula. who woulda thunk that exponentials and sinusoids were so closely related? not so astonished anymore. r b-j
Reply by ●October 11, 20142014-10-11
robert bristow-johnson <rbj@audioimagination.com> wrote:> On Thursday, October 9, 2014 12:08:17 PM UTC-4, Tim Wescott wrote: >> Who was it who figured out that>> integral 1/(x^2 + 1) = arctan x> probably the same person that figured out that> (d/dx) arctan(x) = 1/(x^2 + 1)At some point, yes, the favorite way to make integral tables was to take derivatives of as many different functions as one could think up, and then write down the inverse as an integral. I don't know if that was one, though.> i dunno. i was sorta astonished first time i saw Euler's formula. > who woulda thunk that exponentials and sinusoids were so closely > related? not so astonished anymore.There is a chapter in "Feynman Lectures on Physics" where he derives it, somewhat indirectly. For one, he answers the question of where log tables came from. Take 10 successive square roots of 10, which gives you 10**0.5, 10**0.25, 10**0.125, and so on. That gives you 10 different fractional powers of 10, or, the other way around, 10 entries for a log table. If you multiply them, you can get the rest of the 1024 entries for a log tabl. And finally, you discover that for small enough values linear interpolation works. You find that for small x, 10**x is about 1+2.305*x, or that log10(1+x) is about 0.4338*x. Then he makes an interesting guess. That 10**x is still about 2.303*x when x is small and complex, and so computes 10**(i/1024) as 1+0.00225*i. From that, he computes integer powers of 1+0.00225*i, and graphs the real and imaginary parts. That is, 10**(i*n/1024) for integer values of n. The result is sinusoids with a period of 2.688. Then, asking what base to an imaginary power would generate sinusoids with period 2*pi, he finds that it is e. -- glen
