How to prove the definition of arctangent function using integral?

In summary, the conversation is about a problem from the book "Introduction to Analysis" by Arthur P. Mattuck, specifically chapter 20, problem 20-1. The person has worked out questions (a), (b), and (c), but is stuck on question (d). They are seeking help and a suggestion is given to change the numerator to make it integrable and to compare it to the integral of another function with a limiting value of less than or equal to 2.5 at infinity. It is important to make sure that the comparison function is always larger and some "glue" may be needed.
  • #1
zhongbeyond
3
0

Homework Statement


This is a problem from Introduction to Analysis by Arthur P. Mattuck,chapter 20,problem 20-1.

<a href="http://www.flickr.com/photos/86024731@N04/8090259684/" title="arctangent by gnu is not unix, on Flickr"><img src="http://farm9.staticflickr.com/8193/8090259684_a5ce06801e.jpg" width="500" height="325" alt="arctangent"></a>

I have worked out question (a),(b),(c),but I am stucked in question (d).
Please help,thanks.
 
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  • #2
For d.) you will need to compare the given integral to the integral of some other function which you can show (easily) has limiting value < or = to 2.5 at infinity.
Suggestion: consider changing the numerator to make it integrable. Be sure to check that your comparison function is everywhere bigger. (You may need to use some glue.)
 

Related to How to prove the definition of arctangent function using integral?

1. What is the definition of the arctangent function?

The arctangent function (arctan or tan-1) is the inverse function of the tangent function. It is defined as the angle, measured in radians, whose tangent is equal to a given number.

2. What does the integral representation of the arctangent function mean?

The integral representation of the arctangent function is an alternative way of expressing the function using an integral. It allows us to find the value of the function for any given input by evaluating the integral.

3. How do you prove the definition of the arctangent function using integral?

The proof involves using the properties of integrals and the fundamental theorem of calculus to show that the derivative of the integral representation of the arctangent function is equal to the original function itself.

4. Why is the integral representation of the arctangent function important?

The integral representation of the arctangent function is important because it allows us to evaluate the function for any given input and also provides an alternative expression for the function. It is also useful in solving problems involving the arctangent function in calculus and other areas of mathematics.

5. Are there any real-life applications of the arctangent function and its integral representation?

Yes, the arctangent function and its integral representation have numerous real-life applications in fields such as physics, engineering, and computer science. For example, they are used in calculating the angles of elevation and depression in navigation and in determining the trajectory of projectiles in physics problems.

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