Tanc Function

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

(1)

Since  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 , as introduced here, could be coined, although there appears to be no name previously assigned to this function.

The derivative is given by

(2)

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

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

	OEIS	root
0		0
1	A115365	4.4934094579090641753...
2		7.7252518369377071642...
3		10.904121659428899827...
4		14.066193912831473480...
5		17.220755271930768739...

The positive solutions can be written explicitly in series form as

(3)

(OEIS A079330 and A088989), where the series in  can be found by series reversion of the series for  and

(4)

for  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.

Because of the vertical asymptotes of  as odd multiples of , this function is much less well-behaved than the sinc function, even as . The plot above shows  for integers . The values of  giving incrementally smallest values of are , 11, 1317811389848379909481978463177998812826691414678853402757616, ...(OEIS A079331), corresponding to values of , , , , .... Similarly, the values of  giving incrementally largest values of are , 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 . In addition, since they must be near an odd multiple of  in order for  to be large, the corresponding denominators must be odd. There is also a very strong correlation between  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 ).

smallest	convergent	largest
	1	1.55741
	2
	4
	15	2.659341
	17	3.582052
	19	4.331096
	29	18.007800
	118
	136
	233	18.056613
	315	556.306227
	1134
	1568
	1718
	2154
	2468
	3230
	3727	2750.202396
	3763	10539.847388
	5187
	8872
	9768
	11282
	12284
	15503	24263.751532
	24604
	153396
	156559	228085.415076

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

REFERENCES:

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