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.
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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 of is measured as .
The arc tangent function, denoted or , is a function defined as follows: for , is the unique number in the open interval such that .
|default domain||all real numbers, i.e., all of|
|range|| the open interval , i.e., the set |
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)||(the origin) only|
|horizontal asymptotes|| The line as |
The line as
|derivative||(see #First derivative)|
|second derivative||(see #Second derivative)|
|higher derivatives||The derivative is a rational function of where the denominator degree is more than the numerator degree.|
|antiderivative||. We use integration by parts (see #Integration).|
|higher antiderivatives||The function can be antidifferentiated any number of times in terms of elementary functions.|
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:
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:
WHAT WE USE: chain rule for differentiation and differentiation rule for power functions
The second derivative is given as:
We can integrate this using the inverse function integration method, and obtain:
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 .
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.
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: