# Complex anaylsis, winding number question.

• screwyshrew
In summary, the equation for the winding number/index of a curve uses an arbitrary point ζ on the curve γ as the variable of integration. When parametrizing the curve by an interval [a,b], the integral is taken from a to b. An example of this equation in use is integrating dw/w over the unit circle, parametrized by cos(t) + i sin(t) for the interval [0,1], which gives the winding number about z=0 of the unit circle (multiplied by 2πi).
screwyshrew
So the in the equation for the winding number/index of a curve

I($\gamma$, z) = $\frac{1}{2i\pi}$ $\int\gamma \frac{1}{ζ-z}dζ$

where $\gamma$ : [a, b] → ℂ and z is an arbitrary point not on $\gamma$, what exactly does ζ represent?

It represents an arbitrary point of the curve gamma. I.e. it is a bound or "dummy" variable for any point of gamma. a fuller notation would be the integral over all zeta on gamma. But the fact that it appears after the "d" also tells you it is the variable of integration, and is usual.

This is possibly a dumb question (please bear with me): the integration would be taken from a to be, correct? Could you post an example of this equation in use, please?

yes, after parametrizing the curve gamma by an interval [a.b], then the integral is pulled back to that interval and THEN taken from a to b.

e.g. just integrate dw/w over the unit circle, parametrizing it by cos(t) +i sin(t) for the interval [0,1]. try it. that's the winding number about z = 0 of the unit circle. (times 2pi.i)

## 1. What is complex analysis?

Complex analysis is a branch of mathematics that deals with functions of complex numbers. It involves the study of complex-valued functions, which are functions whose domain and range are both complex numbers. This field of study is important in many areas of mathematics, physics, and engineering.

## 2. What is the winding number question in complex analysis?

The winding number question refers to a problem in complex analysis that involves determining the number of times a function winds around a given point in the complex plane. This concept is closely related to the concept of contour integration and has many applications in physics and engineering.

## 3. What is the Riemann mapping theorem?

The Riemann mapping theorem is a fundamental theorem in complex analysis that states that any simply connected open subset of the complex plane can be mapped conformally onto the unit disk. This theorem has many important applications in the theory of complex functions, including the solution of the Dirichlet problem.

## 4. What is the Cauchy-Riemann equations?

The Cauchy-Riemann equations are a set of partial differential equations that characterize differentiability of complex-valued functions. They state that a complex-valued function is differentiable at a point if and only if it satisfies these equations at that point. These equations play a crucial role in the study of complex analysis.

## 5. What are the applications of complex analysis?

Complex analysis has many applications in mathematics, physics, and engineering. Some specific examples include the use of conformal mappings in fluid dynamics, the use of complex analysis in the solution of Laplace's equation and other partial differential equations, and the use of the residue theorem in calculating certain types of integrals. Complex analysis also has connections to number theory, algebraic geometry, and other branches of mathematics.

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