The name tanc function is used here, by analogy with the sinc function, for the function defined for complex
numbers 
with 
by
 |
(1)
|
Since 
is not a cardinal function, the "analogy"
with the sinc function is one of functional structure,
not mathematical properties.
For 
in this domain,
its derivative is given by
 |
(2)
|
At the origin, 
. 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 nonnegative real roots
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 first positive root can be given in closed form as
 |
(3)
|
where 
is the 
th
positive root of the Bessel
function of the first kind 
.
The positive roots have the series expansion
 |
(4)
|
(OEIS A079330 and A088989), where 
and 
is a positive integer. The series
in 
is obtained by series
reversion of 
about 
(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 
. For positive
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 194000th term of the continued
fraction. All these extrema correspond to numerators
of convergents of the continued
fraction 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.
