Sine integral
From Calculus
This article is about a particular function from a subset of the real numbers to the real numbers. Information about the function, including its domain, range, and key data relating to graphing, differentiation, and integration, is presented in the article.
View a complete list of particular functions on this wiki
For functions involving angles (trigonometric functions, inverse trigonometric functions, etc.) we follow the convention that all angles are measured in radians. Thus, for instance, the angle ofis measured as
.
Contents
Definition
The function, denoted , is defined as the definite integral of the sinc function from 0 to the input point:
Recall that the sinc function is defined in terms of the sine function as follows:
Key data
Item | Value |
---|---|
default domain | all real numbers, i.e., all of ![]() |
range | ![]() ![]() absolute maximum value: ![]() ![]() |
period | none, the function is not periodic |
horizontal asymptotes | ![]() ![]() ![]() ![]() |
local maximum values and points of attainment | local maximum values are attained at all odd positive integer multiples of ![]() ![]() |
local minimum values and points of attainment | local minimum values are attained at all even positive integer multiples of ![]() ![]() |
points of inflection | at ![]() ![]() ![]() ![]() |
important symmetries | the function is an odd function |
derivative] | sinc function ![]() |
second derivative | ![]() ![]() ![]() ![]() |
antiderivative | ![]() |
power series and Taylor series | ![]() ![]() |
Differentiation
First derivative
The first derivative is the sinc function. This follows from the way the sine integral is defined in the first place.
Second derivative
See sinc function#First derivative.
Integration
First antiderivative
WHAT WE USE: integration by parts
We have:
We obtain this using integration by parts as follows. We take as the part to integrate, getting:
This simplifies to the desired answer.