Schwarz Reflection Principle

Let

$$g(z)\equiv \sum_{n=0}^\infty (z-z_0)^n {f^{(n)}(z_0)\over n!},$$ (1)

then

$$g^*(z) = \left[{\,\sum_{n=0}^\infty (z-z_0)^n {f^{(n)}(z_0)\... ...= \sum_{n=0}^\infty (z^*-{z_0}^*)n {f^{(n)}({z_0}^*)\over n!}.$$ (2)

If $z_0$ is pure real, then $z_0 = {z_0}^*$, so

$$g^*(z) = \sum_{n=0}^\infty (z^*-z_0)^n {f^{(n)}(z_0)\over n!} = g(z^*).$$ (3)

Therefore, if a function $f(z)$ is Analytic over some region including the Real Line and $f(z)$ is Real when $z$ is real, then $f^*(z) = f(z^*)$.