The Function
William B. Gearhart
Harris S. Shultz
The College Mathematics Journal
, March 1990, Volume 21, Number 2, pp. 9099.
William B. Gearhart
received his B.S. degree in engineering physics from Cornell
University, and his Ph.D degree in applied mathematics, also from Cornell University.
He is currently Professor of Mathematics at the California State University, Fullerton.
His research interests include approximation theory, numerical analysis, optimization
theory, and mathematical modeling.
Harris S. Shultz
, Professor of Mathematics at California State University, Fullerton, is
the author of more than forty published papers and a textbook,
Mathematical Topics for
Computer Instruction.
He has been CoPrincipal Investigator of the Southern California
Mathematics Honors Institute, the Mariana Islands Mathematics Institute, and the
Teacher Training Project, all funded by the National Science Foundation. In 1988 he
was named Outstanding Professor at California State University, Fullerton, and in 1989
he was the recipient of the California State University Trustees’ Outstanding Professor
Award.
C
alculus students are well aware of an important use of the function sin
The common and perhaps the easiest approach to finding the derivatives of the
trigonometric functions hinges on the result
(1)
However, few students realize that this function plays a key role in many areas of
mathematics and its applications. In this paper, we briefly describe several of these roles
after first presenting some elementary examples in which sin
arises geometrically.
These examples are accessible to calculus students and may help motivate interest in and
draw attention to this function. In addition, they provide some unusual proofs of the
limit (1).
Properties of sin
In certain fields, such as signal analysis, the function sin
is often called the “sinc
function.” Thus, let us set
By defining sinc
the function is extended to an analytic function on the real line.
We shall refer to this extension also by the name sinc. The graph of sinc is shown in
Figure 1, where we note that it is an even function with roots at
for
that
for all
and that lim
sinc
x
5
0.
x
→
±
‘
x
0,

sinc
x

<
1
±
2, . . .,
n
5
±
1,
n
p
,
0
5
1,
sinc
x
5
sin
x
x
.
x
y
x
x
/
x
x
y
x
lim
x
→
0
sin
x
x
5
1.
x
y
x
.
C
3
sin
x
x
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Figure 1
The sinc function has a rapidly convergent power series representation,
and even an infinite product representation [
7
],
Also,
although this integral is not absolutely convergent. However, the square of sinc is
integrable on the real line with
All things considered, sinc is a rather nice, wellbehaved function. Properties of sinc are
presented in [
4
, sections 39–41] and [
2
, pages 62–63].
Examples from Geometry
In this section we present a few simple examples from geometry involving sin
.
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 Math, Calculus, Fourier Series, Periodic function, Mathematical analysis, sinc

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