What is the explicit formula for the antiderivative of a holomorphic function used in this proof?

Conditions

• The function f is holomorphic on the domain D.
• The domain D is simply connected (ensuring path independence).
• A base point $z_0$ is fixed in D.
• $\gamma_z$ is a path in D from $z_0$ to z.

Reasoning, step by step

1. Fix an arbitrary base point $z_0$ inside the domain D.
2. For any point $z \in D$, let $\gamma_z$ be a path lying entirely within D that starts at $z_0$ and ends at $z$.
3. Define the function $F(z)$ as the integral of f along this path: $F(z) := \int_{\gamma_z} f(w) dw$.
4. Verify that this definition is well-defined due to the path independence guaranteed by the simply connected domain and holomorphicity of f.

Example

The board shows the formula $F(z) := \int_{\gamma_z} f(w) dw$, and the speaker explains, 'We can construct such a function F using contour integration... We then define the function $F(z)$ as the integral of f along this path'.

Common misconceptions

• Assuming the antiderivative is defined by an indefinite integral without specifying a base point and path.
• Thinking the formula applies to any continuous function, ignoring the necessity of holomorphicity and a simply connected domain for path independence.

Watch the explanation

Explore next

• Why is the domain D being simply connected important for defining the antiderivative $F(z)$ via a contour integral?
• How does the openness of a domain help in calculating derivatives?