We can use a straight line segment because the function f is holomorphic on a simply connected domain D, which guarantees that the contour integral of f is path-independent. This allows us to strategically choose the simplest path—the straight line segment from a to z—to evaluate the difference F(z)−F(a).
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.
We can use a straight line segment because the function f is holomorphic on a simply connected domain D, which guarantees that the contour integral of f is path-independent. This allows us to strategically choose the simplest path—the straight line segment from a to z—to evaluate the difference F(z)−F(a).
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.
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 z0 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 z0 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 z0 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 z0 to z.