Are there complex functions with finite, nonzero branch points?

In summary, the conversation is about branch points in complex analysis and the difficulty in imagining or coming up with functions that have nonzero branch points. The difficulty lies in the fact that for a point to be considered a branch point, the function's values along any closed path must be different. This is specific for a branch point at zero, as for a branch point at a different point, the closed paths to consider would be those with a winding number around that point. The suggestion is to look for Riemann surfaces in textbooks or online for a conceptual explanation of branching functions. A hint is also given to find the branch points of a specific function.
  • #1
fuserofworlds
12
0
I just completed a brief introduction to branch points in complex analysis, and I find it difficult to imagine/come up with functions with nonzero branch points.

My difficulty is this: for the point to be considered a branch point, f(r,θ) and f(r,θ+2π) must be different for ANY closed path. If the branch point is zero, you can arbitrarily shrink the path and still be sweeping from 0 to 2π. If your branch point is anything but the origin, then as the path becomes arbitrarily small, so does the angle you sweep over. In such a case, how can you get multiple values for the same input?

Thanks in advance for any answers to this problem.
 
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  • #2
What if you just translate a function with branch point at 0?
 
  • #3
fuserofworlds said:
f(r,θ) and f(r,θ+2π) must be different for ANY closed path.
You mean for any closed path with a winding number around zero. This is specific for a branch point at zero. For a branch point somewhere else, the closed paths to consider are those with a winding number around that other point, not around zero.
 
  • #4
fuserofworlds: look for Riemann surfaces in textbooks or the Internet. They provide a conceptual explanation of branching functions.
 
  • #5
Hint: find the branch points of [itex] \sqrt{(z-a)(z-b)}[/itex] for [itex]a \neq b[/itex] and neither [itex]a[/itex] or [itex]b[/itex] are zero.

jason
 

Related to Are there complex functions with finite, nonzero branch points?

1. What is a complex function?

A complex function is a mathematical function that takes a complex number as its input and produces a complex number as its output. Complex numbers are numbers that have both a real and imaginary component.

2. What are branch points in a complex function?

Branch points are points in the complex plane where a function becomes multivalued. This means that there are multiple possible outputs for a single input value, resulting in branches or loops in the function's graph.

3. Can a complex function have finite, nonzero branch points?

Yes, it is possible for a complex function to have finite, nonzero branch points. These points can occur when the function has a finite number of branches or when the branches converge to a finite point.

4. How do branch points affect the behavior of a complex function?

Branch points can significantly impact the behavior of a complex function. They can cause discontinuities, non-analyticity, and non-uniform convergence, making the function more complex and difficult to analyze.

5. Are there any practical applications for complex functions with branch points?

Yes, complex functions with branch points have many practical applications, particularly in physics and engineering. They are used in the study of electromagnetic fields, fluid dynamics, and quantum mechanics, among other fields.

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