Complex analysis - removable singular points

In summary, the conversation discusses the concept of removable singularities and their relation to analytic functions and integrals. It is noted that the residue at a removable singularity is always zero, but this does not necessarily mean the function is analytic. The example of (z-sinz)/z3 is given, which has a removable singularity at z=0 but is still an analytic function. The importance of removing singularities before studying a function is emphasized. Additionally, it is pointed out that a function can be analytic even if it has a removable singularity. The function (z-sinz)/z3 is given as an example, which only converges if f(0) is defined, making it an analytic function.
  • #1
dyn
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Hi. I have 2 questions regarding removable singular points.
1 - the residue at a removable singularity is always zero so by the residue theorem the integral around a closed simple contour is zero. Cauchy's theorem states the integral around a simple closed contour for an analytic function is zero so am I right in thinking that the integral being zero does not imply the function is analytic ? As it is also zero for functions with a removable singularity.

2 - I have seen the following example in a book where it is shown that (z-sinz)/z3 has a removable singularity at z=0 but then it states that the series converges for all values of z but surely the function is undefined at z=0 ?
Thanks
 
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  • #2
1. there is no difference in integrating an analytic function versus a function with a removable singularity. more interesting is the case of a function which does not have a removable singularity but still has integral zero due to there being a pole there of higher order and residue zero. i.e. integral = 0 just means the term 1/z has coefficient zero, but therte could still be terms of order 1/z^2 or 1/z^3 etc with non zero coefficients but the integral will not detect that.

2. in this example sin has only odd degree terms of positive degree, and the linear term equals z, so subtracting off z reduces it to beginning in degree 3. thus dividing by z^3 gives an analytic function (i.e. one with a removable singularity). do not obsess over whether a function has or has not yet been stated as defined at a point when it is perfectly capable of being well defined there. all functions with removable singularities should have them removed, before studying the function.
 
  • #3
Thanks for your reply.
1- so if an integral around a closed loop is zero ; that doesn't tell me if the function is analytic or not ?
2- You say that (z-sinz)/z3 is an analytic function ie. one with a removable singularity. Aren't they 2 different things ? A singular point is a point where a function is not analytic.
 
  • #4
A removable singularity just means you chose a poor way to define a function that could be analytic at that point. Like ##f(z)=\frac{z^2+z}{z}##. It has a removable singularity at z=0. You could have defined f(z)=z+1 and you wouldn't have had that issue.
 
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  • #5
Thanks. So for the function f(z) = (z-sinz)/z3 ; it only converges if f(0) is defined and so it is analytic ? If f(0) is not defined it does not converge and is not analytic ?
 
  • #6
f(z) converges for z->0 which is independent of what happens at exactly 0.

##f(z)=\frac 1 6 - \frac{z^2}{120} + \frac{z^4}{5040}\pm ...##
 

What is a removable singular point in complex analysis?

A removable singular point is a point in a complex function where the function is undefined, but can be made continuous by defining the value of the function at that point. In other words, the function can be extended to include that point without changing the overall behavior of the function.

How can a removable singular point be identified in a complex function?

A removable singular point can be identified by analyzing the behavior of the function around the point. If the function is undefined at the point but remains bounded and continuous in a small neighborhood around the point, then it is a removable singular point.

What is the significance of removable singular points in complex analysis?

Removable singular points have important implications for the behavior and properties of complex functions. They can affect the differentiability, integrability, and convergence of a function, and can also reveal information about the structure of the function.

Can a removable singular point be removed by simply defining the function at that point?

Yes, a removable singular point can be removed by defining the value of the function at that point. This is known as "filling in" the point and results in a continuous function without changing the overall behavior of the function.

Are there other types of singular points in complex analysis besides removable singular points?

Yes, there are other types of singular points, including poles and essential singular points. Poles are points where the function approaches infinity, and essential singular points are points where the function has an essential singularity and cannot be made continuous by defining the value at that point.

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