*Intermediate Physics for Medicine and Biology*, Russ Hobbie and I write

The function sin(Sinc(x)/xhas its maximum value of 1 atx= 0. It is also called the sinc(x) function.

*x*) oscillates like sin(

*x*), but its amplitude decays as 1/

*x*. If sin(

*x*) is zero then sinc(

*x*) is also zero, except for the special point

*x*= 0, where 0/0 becomes 1.

A plot of the sinc function. |

Trigonometric Delights, by Eli Maor |

*IPMB*, Russ and I don’t evaluate the values of

*x*corresponding to local maximum and minimum values of sinc(

*x*). Eli Maor examines the peak values of

*f*(

*x*) = sinc(

*x*) in his book

*Trigonometric Delights*. He writes

We now wish to locate theextreme pointsoff(x)—the points where it assumes its maximum or minimum values. And here a surprise is awaiting us. We know that the extreme points ofg(x) = sinxoccur at all odd multiples of π/2, that is, atx= (2n+1)π/2. So we might expect the same to be true for the extreme points off(x) = (sinx)/x. This, however, is not the case. To find the extreme point, we differentiatef(x) using the quotient rule and equate the result to zero:

f’(x) = (xcosx– sinx)/x^{2}= 0. (1)

Now if a ratio is equal to zero, then the numerator itself must equal to zero, so we havexcosx– sinx= 0, from which we get

tanx=x. (2)

Equation (2) cannot be solved by a closed formula in the same manner as, say, a quadratic equation can; it is atranscendental equationwhose roots can be found graphically as the points of intersection of the graphs ofy=xandy= tanx.

A plot of y=tanx versus x and y=x versus x. |

The extreme values are at

*x*= 0, 4.49 = 1.43π, 7.73 = 2.46π, etc. As

*x*becomes large, the roots approach (2

*n*+1)π/2.

Eli Maor is a rare breed: a writer of mathematics. Russ and I cite his wonderful book

*e, The Story of a Number*in Chapter 2 of

*IPMB*. I also enjoyed

*The Pythagorean Theorem: A 4,000-year History*. Maor has written many books about math and science. His most recent came out in May:

*Music by the Numbers--From Pythagoras to Schoenberg*. I put it on my summer reading list.

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