Inverse of Integration?

In summary: If you try to do the same thing with f(x)=-x^2 inside a circle with center at z_0, you will get a negative number. This is because the integral is not continuous at the point z_0.
  • #1
207
0
Question:

In my homework problem, I'm trying to get rid of an integration over a close curve on the complex plane. I was wondering if its possible to somehow take the inverse of the integral on other side of the following equatin.

Please explain the reasoning of why or why not so that I can understand it for future problems :P.

2ipif'(0) = Integral over alpha (Any circle containing 0 on the interior) of f(x)/x^2.

Where f(x) is an analytic function from c to c.

Obviously f(x)/x^2 is not analytic on all c, since 0 is a critical point, I was still wondering if there is a way to get rid of the integral by doing something on both sides of the equation. Thanks for any help you can provide.
 
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  • #2
Does the integral have bounds? If so, no inverse. Otherwise, differentiate.
 
  • #3
Use the residue theorem.
 
  • #4
moo5003 said:
Question:

In my homework problem, I'm trying to get rid of an integration over a close curve on the complex plane. I was wondering if its possible to somehow take the inverse of the integral on other side of the following equatin.

Please explain the reasoning of why or why not so that I can understand it for future problems :P.

2ipif'(0) = Integral over alpha (Any circle containing 0 on the interior) of f(x)/x^2.

Where f(x) is an analytic function from c to c.

Obviously f(x)/x^2 is not analytic on all c, since 0 is a critical point, I was still wondering if there is a way to get rid of the integral by doing something on both sides of the equation. Thanks for any help you can provide.

What you have is a special case of Cauchy's integral formula:
[tex]2\pi i f^{(n)}(z_0)= \int \frac{f(z)}{(z-z_0)^{n+1}}dz[/tex]
where the integration is over any closed path (not necessarily a circle) with z0[/sup] in its interior and f(z) is analytic inside the path. You can prove it by writing f as a Taylor's series, dividing each term by (z- z0)n+1 and then integrating term by term. It is easy to show that [itex]\int (z-z_0)^n dz[/itex] over a closed path containing z0 is 0 for any n other than -1 and is [itex]2\pi i[/itex] when n= -1.
 

What is the inverse of integration?

The inverse of integration, also known as antidifferentiation, is the process of finding a function that, when differentiated, gives the original function. It is the reverse process of integration.

What is the difference between integration and inverse of integration?

Integration is the process of finding the area under a curve, while inverse of integration is the process of finding the function that gives the original curve when differentiated. In other words, integration is like finding the sum of small parts, while inverse of integration is like finding the individual parts.

What are the methods for finding the inverse of integration?

There are several methods for finding the inverse of integration, including substitution, integration by parts, partial fractions, and trigonometric substitution. The choice of method depends on the form of the original function.

What are the practical applications of the inverse of integration?

The inverse of integration has many practical applications in fields such as physics, engineering, and economics. It is used to find the position, velocity, and acceleration of objects in motion, as well as to solve problems involving rates of change and optimization.

Is it possible for a function to have multiple inverses of integration?

Yes, it is possible for a function to have multiple inverses of integration. This is because the process of integration involves adding a constant term, which can result in different functions with the same derivative. However, these functions will only differ by a constant, so they are considered equivalent.

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