Theorem connecting the inverse of a holomorphic function to a contour integral

In summary, a holomorphic function is a complex-valued function that is differentiable at every point in its domain and can be represented by a power series. The inverse of a holomorphic function is another holomorphic function that "undoes" the original function and can be expressed as a contour integral. This connection is known as the Cauchy integral formula, which allows us to calculate the values of holomorphic functions at points inside their domains. However, there are limitations to this theorem, including the function being holomorphic in a simply connected region and the contour being a closed curve that does not intersect itself.
  • #1
IDontKnow1
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I tried posting this at stack exchange but it never got the question answered. I want to prove this:

If f:U→C is holomorphic in U and invertible, P∈U and if D(P,r) is a sufficently small disc about P, then


[itex]f^{-1}(w) = \frac{1}{2\pi i} \oint_{\partial D(P,r)}{\frac{sf'(s)}{f(s)-w}}ds[/itex]
 
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  • #2
That is just Cauchy's formula.

[tex]f^{-1}(w)=\frac{1}{2\pi\imath}\oint_{\partial D(P,r)} \frac{s}{f(s)-w}d(f(s))[/tex]
 

1. What is a holomorphic function?

A holomorphic function, also known as an analytic function, is a complex-valued function that is differentiable at every point in its domain. This means that it has a well-defined derivative at every point and can be represented by a power series in its domain.

2. What is the inverse of a holomorphic function?

The inverse of a holomorphic function is another holomorphic function that, when composed with the original function, gives the identity function. In other words, it "undoes" the original function.

3. How is the inverse of a holomorphic function connected to a contour integral?

The inverse of a holomorphic function can be expressed as a contour integral, which is a type of line integral in the complex plane. This connection is known as the Cauchy integral formula, which states that the value of a holomorphic function at a point is equal to the average value of the function around a closed contour enclosing that point.

4. What is the significance of the Cauchy integral formula?

The Cauchy integral formula is significant because it allows us to calculate the values of holomorphic functions at points inside their domains, by using information about the values of the function on the boundary of the domain. This is useful in many areas of mathematics and physics, including complex analysis and potential theory.

5. Are there any limitations to the theorem connecting the inverse of a holomorphic function to a contour integral?

Yes, there are some limitations to this theorem. The function must be holomorphic in a simply connected region, meaning that it cannot have any holes or branch cuts in its domain. Additionally, the contour must be a closed curve that does not intersect itself. If these conditions are not met, the theorem may not hold.

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