**Calculus of the Sinc Function**

**Introduction and Setup**

The unnormalized Sinc function is defined as:

unsinc(x) = sin x / x

And the normalized Sinc function is defined as:

sinc(x) = sin( π x ) / ( π x )

Two things to assume about the sinc function:

1. The function is not defined at x = 0, and

2. The function uses radian angle measure.

(x ≠ 0, and assume radians measure)

Let α be a real constant and define f(x) as:

f(x) = sin( α x ) / (α x)

When α = 1, f(x) becomes unsinc(x). Likewise, when α = π, f(x) becomes sinc(x). I am going to analyze this function f(x).

**Limit**

Limit of sin( α x ) / ( α x )

lim x → 0 ( sin( α x ) / ( α x ) ) → sin 0 / 0 → 0 / 0

This form of improper limit allows us to use L'Hôspital's Rule, which allows us to take the derivative of both the numerator function and denominator function:

lim x → 0 ( α * cos ( α x ) / α ) → lim x → 0 ( cos ( α x ) ) → cos ( 0 ) → 1

Hence lim x → 0 ( sin( α x ) / ( α x ) ) → 1

**Derivative**

Taking the derivative will call for use to use the quotient rule:

d/dx [n(x) / d(x)] = ( n '(x) * d(x) - n(x) * d '(x)) / (d^2(x))

Then:

d/dx [ sin( α x ) / ( α x ) ]:

n(x) = sin ( α x )

n'(x) = α * cos( α x )

d(x) = α * x

d^2(x) = (α * x)^2

d'(x) = α

d/dx [ sin( α x ) / ( α x ) ]

= [ α * cos( α x ) * α * x - sin( α x ) * α ] / [ α^2 * x^2 ]

= [ α^2 * cos( α x ) * x - sin( α x ) * α ] / [ α^2 * x^2 ]

= [ α * cos( α x ) * x - sin( α x ) ] / [ α * x^2 ]

= cos( α x ) / x^2 - sin( α x ) / (α * x^2)

**Integral**

The integral of

∫ sin( α x ) / ( α x ) dx

does not look like it can easily integrated.

Let's use the Taylor Series approach:

sin x = x - x^3 / 3! + x^5 / 5! - x^7 / 7! + x^9 / 9! + . . .

sin( α x ) = ( α x ) - ( α x )^3 / 3! + ( α x )^5 / 5! - ( α x )^7 / 7! + ( α x )^9 / 9! + ...

With x≠0

sin( α x ) / ( α x )

= 1 - ( α x )^2 / 3! + ( α x )^4 / 5! - ( α x )^6 / 7! + ( α x )^8 / 9! + ...

= 1 - α^2 * x^2 / 3! + α^4 * x^4 / 5! - α^6 * x^6 / 7! + α^8 * x^8 / 9! + ...

Now integrate the series:

∫ sin( α x ) / ( α x ) dx

= x - ( α^2 * x^3 ) / (3 * 3!) + ( α^4 * x^5 ) / (5 * 5!) - ( α^6 * x^7 ) / (7 * 7!) + ( α^8 * x^9 ) / (9 * 9!) + ... + C

= x - ( α^2 * x^3 ) / 18 + ( α^4 * x^5 ) / 600 - ( α^6 * x^7 ) / 35280 + ( α^8 * x^9 ) / 3265920 + ... + C

Eddie

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