Why can we use a straight line segment to calculate $F(z) - F(a)$ in the difference quotient?

Conditions

• The function f is holomorphic on the domain D.
• The domain D is simply connected.
• The points a and z are in D, and the straight line segment connecting them lies entirely within D.

Reasoning, step by step

1. Recall the definition of $F(z)$ as the integral of f from a fixed base point $z_0$ to z.
2. Express the difference $F(z) - F(a)$ as the integral of f from a to z.
3. Apply the property of path independence: since f is holomorphic on a simply connected domain, the integral from a to z is the same along any path.
4. Choose the straight line segment $\gamma_z$ from a to z as the convenient path for integration.
5. Substitute the integral over this straight line segment into the difference quotient expression.

Example

The lecturer states, 'as I have told you a minute ago, it doesn't matter which way you choose. So in particular you can choose the following way... going from z zero to a and then to z.' The diagram shows a straight line segment connecting a and z.

Common misconceptions

• Assuming that one can always arbitrarily choose a convenient path to evaluate contour integrals or define antiderivatives without considering the domain's topology and the function's analyticity.
• Thinking that the straight line segment is the only valid path, rather than just a convenient choice due to path independence.

Watch the explanation

Explore next

• How is the estimation lemma applied to bound the integral of $f(w) - f(a)$?
• What is the explicit formula for the antiderivative of a holomorphic function used in this proof?