Vtyjkyuk

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Ch5.2

2 Questions about Cor 5.8 and Cor 5.9 (*)

Question 1. Can we prove Cor 5.9 using Cor 5.8?

Question 2. Can we prove Cor 5.9 without using Cor 5.8?

My proof for either starts out the same:

Consider two paths $gamma_1, gamma_2 subset G$ that are piecewise smooth and have the same start and end points. Denote $-gamma_2 subset G$ as $gamma_2$ passed in the reverse direction. Denote $gamma_1 wedge -gamma_2 subset G$ as the path that starts at the start of both $gamma_1$ and $gamma_2$ and passes $gamma_1$ until the end of both $gamma_1$ and $gamma_2$, w/c is equivalent to the start of $-gamma_2$ and then passes $-gamma_2$ until the end of $-gamma_2$, w/c is equivalent to the start of both $gamma_1$ and $gamma_2$. Observe that $gamma_1 wedge -gamma_2 subset G$ is a closed and piecewise smooth path and thus

$$0 stackrel(**)= int_gamma_1 wedge -gamma_2 f := int_gamma_1 f + int_-gamma_2 f := int_gamma_1 f - int_gamma_2 f implies int_gamma_1 f = int_gamma_2 f$$

This shows that $forall gamma subset G$ piecewise smooth, $int_gamma f$ has the same value because $forall gamma_1, gamma_2 subset G$ piecewise smooth with the same start and end points as $gamma$, $int_gamma_1 f = int_gamma_2 f$.

$$therefore, int_gamma f textis path independent.$$

QED

(**) The justification for this I believe can be done using or without using Cor 5.8. I moved my attempts to an answer.

(*)

(Cor 5.8) If $f$ is holomorphic function on a simply-connected region $G subseteq mathbb C$, then $f$ has an antiderivative.

(Cor 5.9) If $f$ is a holomorphic function on a simply-connected region $G subseteq mathbb C$, then $forall gamma subset G$ piecewise smooth, $int_gamma f$ is path independent.

Question 1.

Pf of Cor 5.9 using Cor 5.8:

Consider two paths $gamma_1, gamma_2 subset G$ that are piecewise smooth and have the same start and end points. Denote $-gamma_2 subset G$ as $gamma_2$ passed in the reverse direction. Denote $gamma_1 wedge -gamma_2 subset G$ as the path that starts at the start of both $gamma_1$ and $gamma_2$ and passes $gamma_1$ until the end of both $gamma_1$ and $gamma_2$, w/c is equivalent to the start of $-gamma_2$ and then passes $-gamma_2$ until the end of $-gamma_2$, w/c is equivalent to the start of both $gamma_1$ and $gamma_2$. Observe that $gamma_1 wedge -gamma_2 subset G$ is a closed and piecewise smooth path and thus by $colorblue(1)$ below,

$$0 stackrelcolorblue(1)= int_gamma_1 wedge -gamma_2 f := int_gamma_1 f + int_-gamma_2 f := int_gamma_1 f - int_gamma_2 f implies int_gamma_1 f = int_gamma_2 f$$

This shows that $forall gamma subset G$ piecewise smooth, $int_gamma f$ has the same value because $forall gamma_1, gamma_2 subset G$ piecewise smooth with the same start and end points as $gamma$, $int_gamma_1 f = int_gamma_2 f$.

$$therefore, int_gamma f textis path independent.$$

QED Cor 5.9 using Cor 5.8

Pf $colorblue(1)$:

We can use a corollary (Cor 4.13) to the complex analogue of the Fundamental Theorem of Calculus Part II (Thm 4.11) because

• $f$ has an antiderivative on $G$ by Cor 5.8, w/c we can apply $because f$ is holomorphic on a simply-connected region, by assumption




• $f$ is continuous on an open subset $G because f$ is holomorphic on $G$ by assumption,




• $G$ is open $because G$ is a simply-connected region by assumption and




• $gamma_1 wedge -gamma_2$ is piecewise smooth and closed $because gamma_1$ and $ -gamma_2$ are too $because gamma_1$ and $gamma_2$ are too.




• Antiderivatives can be defined on open disconnected subsets and need not be only for regions as the book defines.

QED $colorblue(1)$

Question 2.

Pf of Cor 5.9 without using Cor 5.8:

Consider two paths $gamma_1, gamma_2 subset G$ that are piecewise smooth and have the same start and end points. Denote $-gamma_2 subset G$ as $gamma_2$ passed in the reverse direction. Denote $gamma_1 wedge -gamma_2 subset G$ as the path that starts at the start of both $gamma_1$ and $gamma_2$ and passes $gamma_1$ until the end of both $gamma_1$ and $gamma_2$, w/c is equivalent to the start of $-gamma_2$ and then passes $-gamma_2$ until the end of $-gamma_2$, w/c is equivalent to the start of both $gamma_1$ and $gamma_2$. Observe that $gamma_1 wedge -gamma_2 subset G$ is a closed and piecewise smooth path and thus by $colorred(2)$ below,

$$0 stackrelcolorred(2)= int_gamma_1 wedge -gamma_2 f := int_gamma_1 f + int_-gamma_2 f := int_gamma_1 f - int_gamma_2 f implies int_gamma_1 f = int_gamma_2 f$$

This shows that $forall gamma subset G$ piecewise smooth, $int_gamma f$ has the same value because $forall gamma_1, gamma_2 subset G$ piecewise smooth with the same start and end points as $gamma$, $int_gamma_1 f = int_gamma_2 f$.

$$therefore, int_gamma f textis path independent.$$

QED Cor 5.9 without using Cor 5.8

Pf $colorred(2)$:

Use a corollary (Cor 4.20) to Cauchy's Thm (Thm 4.18), w/c we can apply because:

• $G$ is a region $because G$ is a simply-connected region,




• $gamma_1 wedge -gamma_2 sim_G 0 because gamma_1 wedge -gamma_2$ is closed, and closed paths are $G$-contractible if $G$ is a simply-connected region, by definition of a simply-connected region and




• $gamma_1 wedge -gamma_2$ is piecewise smooth and closed $because gamma_1$ and $ -gamma_2$ are too $because gamma_1$ and $ gamma_2$ are too.

QED $colorred(2)$