[math-fun] Does this define sin(z)?
1) sin(0) := 0 and 2) |sin(x) - sin(y)| ≤ |x - y| and 3) sin(x) = sin(x/3) (3 - 4 sin(x/3)^2). Note that 2) isn't even true, e.g. x=0, y=i. If this means "No", then what about for real z? —rwg
I spent about a minute thinking about why (1) was necessary. It's because S = S (3 - 4S^2) has more than one root. So that makes me wonder what happens when you substitute (1') s(0) = sqrt(3)/2. Also: even as stated I don't think this suffices, because, for example, the function f(x) = sin(|x|) would also satisfy the functional equation. (That is, (3) never samples the other side of the origin.) On Fri, Dec 13, 2019 at 9:06 AM Bill Gosper <billgosper@gmail.com> wrote:
1) sin(0) := 0 and
2) |sin(x) - sin(y)| ≤ |x - y| and
3) sin(x) = sin(x/3) (3 - 4 sin(x/3)^2).
Note that 2) isn't even true, e.g. x=0, y=i. If this means "No", then what about for real z? —rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I made an algebra error. (1') should be s(0) = sqrt(2)/2. On Fri, Dec 13, 2019 at 9:25 AM Allan Wechsler <acwacw@gmail.com> wrote:
I spent about a minute thinking about why (1) was necessary. It's because S = S (3 - 4S^2) has more than one root. So that makes me wonder what happens when you substitute (1') s(0) = sqrt(3)/2.
Also: even as stated I don't think this suffices, because, for example, the function f(x) = sin(|x|) would also satisfy the functional equation. (That is, (3) never samples the other side of the origin.)
On Fri, Dec 13, 2019 at 9:06 AM Bill Gosper <billgosper@gmail.com> wrote:
1) sin(0) := 0 and
2) |sin(x) - sin(y)| ≤ |x - y| and
3) sin(x) = sin(x/3) (3 - 4 sin(x/3)^2).
Note that 2) isn't even true, e.g. x=0, y=i. If this means "No", then what about for real z? —rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Wouldn't a constant sin(z) = 0 satisfy these conditions? Tom Bill Gosper writes:
1) sin(0) := 0 and
2) |sin(x) - sin(y)| ≤ |x - y| and
3) sin(x) = sin(x/3) (3 - 4 sin(x/3)^2).
Note that 2) isn't even true, e.g. x=0, y=i. If this means "No", then what about for real z? —rwg
Answer: No! sin(z/2) also satisfies this. E.g., In[259]:= sin[x_] := x/2 /; x (1 - x^2/6) == x (* The /2 is bogus! *) In[260]:= sin[x_] := # (3 - 4 #^2) &@sin[x/3] In[262]:= sin[π/6.`22] Out[262]= 0.2588190451025207623491 In[263]:= N[(-1 + Sqrt[3])/(2 Sqrt[2]), 22] Out[263]= 0.2588190451025207623489 In[264]:= ArcSin[(-1 + Sqrt[3])/(2 Sqrt[2])] // FullSimplify Out[264]= π/12 In[220]:= sin[π/3.] Out[220]= 0.500000000000001 But In[259]:= sin[x_] := x /; x (1 - x^2/6) == x (instead of x/2/;...) gives a numerically decent sin function. Tom Karzes (replacing x/2 by x/∞): Wouldn't a constant sin(z) = 0 satisfy these conditions? —————— Allan Wechsler: I spent about a minute thinking about why (1) was necessary. It's because S = S (3 - 4S^2) has more than one root. So that makes me wonder what happens when you substitute (1') s(0) = sqrt(2)/2. Also: even as stated I don't think this suffices, because, for example, the function f(x) = sin(|x|) would also satisfy the functional equation. (That is, (3) never samples the other side of the origin.) —————————— So, not even for real z. —rwg On Fri, Dec 13, 2019 at 6:04 AM Bill Gosper <billgosper@gmail.com> wrote:
1) sin(0) := 0 and
2) |sin(x) - sin(y)| ≤ |x - y| and
3) sin(x) = sin(x/3) (3 - 4 sin(x/3)^2).
Note that 2) isn't even true, e.g. x=0, y=i. If this means "No", then what about for real z? —rwg
Sin(x) = x + O(x^3), Cos(x) = 1 - (1/2)*x^2 + O(x^4); Along with angle doubling rules: http://mathworld.wolfram.com/Double-AngleFormulas.html Does this define Sin(x) and Cos(x)??? —Brad
On Dec 13, 2019, at 8:06 AM, Bill Gosper <billgosper@gmail.com> wrote:
1) sin(0) := 0 and
2) |sin(x) - sin(y)| ≤ |x - y| and
3) sin(x) = sin(x/3) (3 - 4 sin(x/3)^2).
Note that 2) isn't even true, e.g. x=0, y=i. If this means "No", then what about for real z? —rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (4)
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Allan Wechsler -
Bill Gosper -
Brad Klee -
Tom Karzes