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The simply connected property of the domain D is crucial because it guarantees path independence for the contour integral of a holomorphic function. This ensures that the value of the integral from a fixed base point $z_0$ to a variable point $z$ depends only on the endpoints, making the definition of the antiderivative $F(z)$ rigorous and well-defined.

Conditions: The domain D is simply connected.; The function f is holomorphic on D.; The antiderivative $F(z)$ is defined as a contour integral from a fixed base point $z_0$ to z.

The simply connected property of the domain D is crucial because it guarantees path independence for the contour integral of a holomorphic function. This ensures that the value of the integral from a fixed base point $z_0$ to a variable point $z$ depends only on the endpoints, making the definition of the antiderivative $F(z)$ rigorous and well-defined.

Conditions: The domain D is simply connected.; The function f is holomorphic on D.; The antiderivative $F(z)$ is defined as a contour integral from a fixed base point $z_0$ to z.