Arc tangent function
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 arc tangent function, denoted or
, is a function defined as follows: for
,
is the unique number
in the open interval
such that
.
Equivalently, the arc tangent function is the inverse function to the restriction of the tangent function to the interval .
Key data
Item | Value |
---|---|
default domain | all real numbers, i.e., all of ![]() |
range | the open interval ![]() ![]() no absolute minimum or absolute maximum, because the extremes are attained asymptotically at infinity. |
local minimum values and points of attainment | no local minimum values |
local maximum values and points of attainment | no local maximum values |
points of inflection (both coordinates) | ![]() |
horizontal asymptotes | The line ![]() ![]() The line ![]() ![]() |
derivative | ![]() |
second derivative | ![]() |
higher derivatives | The ![]() ![]() ![]() |
antiderivative | ![]() |
higher antiderivatives | The function can be antidifferentiated any number of times in terms of elementary functions. |
Graph
Here is a graph of from a zoomed out position, where the horizontal asymptotes are clear.
Here is a more close-up version, with the tangent line through the origin (the line ) drawn, indicating that the origin is a point of inflection for the graph:
Differentiation
First derivative
WHAT WE USE: inverse function theorem and tangent function#First derivative, which in turn relies on sine function#First derivative, cosine function#First derivative, and quotient rule for differentiation
We use the inverse function theorem, and the fact that the derivative of is
.
By the inverse function theorem, we have:
If , then
and we get:
Plugging this into the above, we get:
Second derivative
WHAT WE USE: chain rule for differentiation and differentiation rule for power functions
The second derivative is given as:
Higher derivatives
For higher derivatives, we use the quotient rule for differentiation, combined with the chain rule for differentiation to deal with powers of .
Integration
First antiderivative
We can integrate this using the inverse function integration method, and obtain:
This becomes:
We have , and we get:
More explicitly, we can do the integration using integration by parts taking as the part to differentiate and
as the part to integrate:
For the second integration, we integrate using the formulation to get
.
Higher antiderivatives
The function can be antidifferentiated any number of times using integration by parts. The reason for this is that the derivative of the function is a rational function, and rational functions can be repeatedly integrated within elementarily expressible functions.
All the antiderivatives can be expressed in the form:
where are polynomial. Note that
is ambiguous up to addition of polynomials of degree
if we are integrating
times.
Higher antiderivatives
The function can be antidifferentiated any number of times using integration by parts.
Power series and Taylor series
Computation of power series
The power series for the function about 0 can be obtained as follows.
We know that for the function , we have the power series:
Integrating with a definite integral, we get:
The left side is , so we get:
By the alternating series theorem, we note that the power series on the right converges for and for
, so by Abel's theorem, we conclude that it converges to
of the respective inputs. We thus get: