How to prove $\int_Cf(z)dz=\int_{C_1}f(z)dz+\int_{C_2}f(z)dz$ for complex integration?

Question

I would like to prove a very simple theory statement as follows:

The conditions:

1. Assume a curve that has length within complex plane $C\subset\mathbb{C}$
2. Assume partitions of $C$ which are $C_1$ and $C_2$ and satisfy $C_1\cup C_2=C\land C_1\cap C_2=\varnothing$
3. Assume a complex function $f:\mathbb{C}\to\mathbb{C}$ that has limit on $C$

The conclusion:

1. If $\int_Cf(z)dz$ exists, then $\int_{C_1}f(z)dz$ and $\int_{C_2}f(z)dz$ exist
2. If $\int_{C_1}f(z)dz$ and $\int_{C_2}f(z)dz$ exist, then $\int_Cf(z)dz$ exists

It is a preassumed property of the complex integration, but I would like to have a formal proof of it so I can really understand it.

Answer 1

Suppose $f$ has a primitive $F$ on some domain containing $C$,

Let us label the endpoints of the curves:

• $C$ has endpoints $C_a, C_b$
• $C_1$ has endpoints $C_a, C_m$
• $C_2$ has endpoints $C_m, C_b$

$$\begin{eqnarray} \int_Cf(z)dz &=& F(C_b) - F(C_a) \\ \int_{C_1}f(z)dz+\int_{C_2}f(z)dz &=& F(C_m) - F(C_a) + F(C_b) - F(C_m) \end{eqnarray} $$

so these integrals are equal.