On Friday, September 20, 2024 at 1:19:38 AM UTC-6 Alan Grayson wrote:
On Friday, September 20, 2024 at 12:47:13 AM UTC-6 Alan Grayson wrote: On Wednesday, September 18, 2024 at 6:50:53 PM UTC-6 Brent Meeker wrote: On 9/18/2024 5:19 AM, John Clark wrote: On Wed, Sep 18, 2024 at 8:12 AM Alan Grayson <[email protected]> wrote: On Wednesday, September 18, 2024 at 5:40:42 AM UTC-6 John Clark wrote: On Wed, Sep 18, 2024 at 1:16 AM Alan Grayson <[email protected]> wrote: *I'll get back to you on this. I was thinking, as x increases positively or negatively, the y values (angles) repeat multiple times, making the function many-to-one. In this case, we're mapping all the real numbers, to a subset of the y-axis. Am I mistaken? AG * *Arctan(1) = the angle whose tangent = 1. Isn't this angle 90 deg or pi/2? So your plot seems wrong, but it's what is on the Internet. AG * *That's wrong. Arctan(1) = pi/4, which is what the plot indicates. But I still think the plot keeps repeating as x increases or decreases. AG* [image: image.png] *1) **The range of the Arctangent function is the interval (-π/2,π/2) and its range is all the real numbers.* * 2) By dividing by π, the range scales to (-1/2, 1/2).* * 3) Adding 1/2 shifts the range to (0,1) * *4) Thus for every real number x there is a unique number y between zero and one that corresponds to it, and that number is Y=1/2 + 1/π Arctan(x) . As I said before, the domain is all the real numbers and the range is (0,1)* *> Yes, but initially you were seeking a 1-1 function, but this one is many-to-one. AG * FOR DARWIN'S SAKE! I GIVE UP! Could'a told ya. Brent *I spoke to a friend who is Emeritus Professor in Mathematics at Cal Poly Pomona. He says the inverse tangent function is MULTI VALUED. AG* *Is the arctan periodic, or multi-valued? Internet answer:* No, the arctangent (arctan) function is not periodic; it is considered a one-to-one function * because its domain is restricted to an interval where the tangent function (which is periodic) is one-to-one, typically from -π/2 to π/2, ensuring that each output value corresponds to a unique input value. * Explanation: - Tangent periodicity: The tangent function (tan(x)) is periodic with a period of π, meaning its values repeat every π radians. - Restricting the domain: To create an inverse function (arctan), *we need to restrict the domain *of the tangent function to an interval where it is not repeating, like (-π/2, π/2). - *By restricting the domain in this way, the arctangent function is no longer periodic. * *IOW, the arctan is single-valued if its domain is restricted to (**-π/2, π/2)*, but Clark defines its domain to all real numbers, making the arctan periodic and thus multi-valued. AG This above is not quite right. Clark is using just the *principle branch *of the tangent function, to get its inverse, which is 1-1. But there are many, in fact an infinite number of branches and this is where the multi-value problem originates. AG -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/9be7fa8d-2447-49f1-8ada-b4498b979787n%40googlegroups.com.
