Antiderivative simply connected regionWhy do analytic functions always have an antiderivative on a simply connected region?

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kubistiedt · Accepted Answer

Step 1Cauchy&#039;s Theorem tells us that             ∫                        γ          1                      f    (    x    )    d    z    =                  ∫                              γ            2                              f      (      x      )      d      z       whenever       γ    1        γ    2   are homotopic, simple curves sharing the same endpoints in f&#039;s domain of analyticity   Ω  .. We can use this fact to define the antiderivative of f in   Ω as follows:   F  (  ω  )  =            ∫              γ              f    (    z    )    d    z    ,  , where   γ is a contour connecting a fixed point       z    0    ∈  Ω to   ω  ..Step 2Analyticity is required simply so that F is well-defined (in general it is not.) Using a parametrization   ρ  (  t  )  =      ω    0    +  (  ω  −      ω    0    )  t  ,  0  ≤  t  ≤  1  ,, you can show that the difference quotient satisfies             F      (      ω      )      −      F      (              ω        0            )              ω      −              ω        0              =            ∫      0      1        f    (          ω      0        +    t    (    ω    −          ω      0        )    )    d    t    .  .

Antiderivative simply connected region Why do analytic functions always have an

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