Analytic curves

Question

I am learning basics of complex analysis and I do not understand what is meant by complex valued real analytic function. I have the following questions

(1) What is a complex valued real analytic function?

(2) What is a simple closed analytic curve?

(3) What is the difference between simple closed curve and simple closed analytic curve?

(4) An example of simple closed analytic curve.

I am confused with this terms. Thank you.

Answer 1

1. A complex-valued real analytic function is a function $f : D \to \Bbb C$, where $D$ is an open subset of $\Bbb R$, such that at any point $x_0 \in D$, there is a neighborhood of $x_0$ such that $f(x) = \sum_{n=0}^\infty c_i(x - x_0)^n$ for all $x$ in the neighborhood. Because the codomain of $f$ is $\Bbb C$, the coefficients $c_i$ can be complex numbers. Note that any such $f$ is the restriction of some complex analytic function to the real domain $D$.
2. , 3.

• A "curve" in $\Bbb C$ is just a continuous function $\phi$ from some open or closed interval $D$ into $\Bbb C$.
• The curve is closed if its domain is a closed interval $D = [a,b]$ and $\phi(a) = \phi(b)$. I.e., the curve is a loop.
• The curve is simple if for all $x,y\in D$ other than $x = a, y = b$ or vice versa, $\phi(x) \ne \phi(y)$. I.e. the curve does not intersect itself.
• Finally, you have "analytic". There is such a concept - it just requires that $\phi$ be analytic in the interior, as discussed in (1). For a closed curve, $\phi$ is required to be extendable beyond $a,b$ as an analytic function, and still have $\phi(x) = \phi(x+b-a)$ on some neighborhood of $a$. But I find it doubtful that you have actually encountered this requirement on any curves while studying basic Complex Analysis. This is a strong restriction on the curve, and it is seldom necessary. All the main results of Complex Analysis are proven for curves which are just continuous. I bet if you were to go back to where you've seen this used and read it carefully, you would see that "analytic" was not used to describe the curve, but rather some complex function.

4. Simple closed curves abound. A square, for instance. A circle is an example of an analytic simple closed curve.