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Tanc Function


TancTancReImTancContours

By analogy with the sinc function, define the tanc function by

 tanc(z)={(tanz)/z   for z!=0; 1   for z=0.
(1)

Since tanz/z is not a cardinal function, the "analogy" with the sinc function is one of functional structure, not mathematical properties. It is quite possible that a better term than tanc(z), as introduced here, could be coined, although there appears to be no name previously assigned to this function.

The derivative is given by

 (dtanc(z))/(dz)=(sec^2z)/z-(tanz)/(z^2).
(2)

The indefinite integral can apparently not be done in closed form in terms of conventionally defined functions.

TancRoots

This function commonly arises in problems in physics, where it is desired to determine values of x for which tanx=x, i.e., tanc(x)=1. This is a transcendental equation whose first few solutions are given in the following table and illustrated above.

nOEISroot
00
1A1153654.4934094579090641753...
27.7252518369377071642...
310.904121659428899827...
414.066193912831473480...
517.220755271930768739...

The first of these solutions can be given in closed form as

 r_1=j_(3/2,1),
(3)

where j_(n,k) is the kth positive root of the Bessel function of the first kind J_n(x).

The positive solutions can be written explicitly in series form as

 x=q-q^(-1)-2/3q^(-3)-(13)/(15)q^(-5)-(146)/(105)q^(-7)-...
(4)

(OEIS A079330 and A088989), where the series in q^(-1) can be found by series reversion of the series for x+cotx and

 q=1/2(2k+1)pi
(5)

for k a positive integer (D. W. Cantrell, pers. comm., Jan. 3, 2003). In practice, the first three terms of the series often suffice for obtaining approximate solutions.

TancIntegers

Because of the vertical asymptotes of tanx as odd multiples of pi/2, this function is much less well-behaved than the sinc function, even as x->infty. The plot above shows tanc(n) for integers n. The values of n giving incrementally smallest values of tanc(n) are n=2, 11, 1317811389848379909481978463177998812826691414678853402757616, ...(OEIS A079331), corresponding to values of -1.09252, -20.541, -54.5197, -74.7721, .... Similarly, the values of n giving incrementally largest values of tanc(n) are n=1, 122925461, 534483448, 3083975227, 214112296674652, ... (OEIS A079332), corresponding to 1.55741, 2.65934, 3.58205, 4.3311, 18.0078, 18.0566, 556.306, ... (D. W. Cantrell, pers. comm., Jan. 3, 2002). The following table (P. Carmody, pers. comm., Nov. 21, 2003) extends these results up through the 194,000 term of the continued fraction. All these extrema correspond to numerators of the continued fraction expansion of pi/2. In addition, since they must be near an odd multiple of pi/2 in order for tanx to be large, the corresponding denominators must be odd. There is also a very strong correlation between tanc(n) and the value of the subsequent term in the continued fraction expansion (i.e., a high value there implies the prior convergent was a good approximation to pi/2).

smallestconvergentlargest
11.55741
-1.092522
-20.5414
152.659341
173.582052
194.331096
2918.007800
-54.519653118
-74.772130136
23318.056613
315556.306227
-92.5732001134
-103.1601921568
-121.3453091718
-155.4449472154
-246.7448102468
-415.8048753230
37272750.202396
376310539.847388
-529.4461265187
-829.7124898872
-958.0071339768
-2534.64559911282
-5430.63461112284
1550324263.751532
-12702.23825724604
-43181.130288153396
156559228085.415076

The sequences of maxima and minima are almost certainly unbounded, but it is not known how to prove this fact.


See also

du Bois-Reymond Constants, Sinc Function, Sinhc Function, Tangent

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References

Sloane, N. J. A. Sequences A079330, A088989, and A115365 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Tanc Function

Cite this as:

Weisstein, Eric W. "Tanc Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TancFunction.html

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