Esther Hoffman

2022-04-30

State an antiderivative $F:D\u27f6\mathbb{C}$ or explain why an antiderivative does not exist.

State an antiderivative $F:D\u27f6\mathbb{C}$ or explain why an antiderivative does not exist.

a) $f(z)={z}^{2},D=\mathbb{C}$

b) $f(z)=z\mathrm{sin}z,D=\mathbb{C}$

c) $f(z)=|z{|}^{2},D=\mathbb{C}$

d) $f(z)=\overline{z},D=\mathbb{C}$

For a) and b) I just performed simple integration for the equations

a) ${z}^{3}/3+C$ and b) $-z\mathrm{cos}z+\mathrm{sin}z+C$ but I am stuck on c) and d). It seems like an antiderivative does not exist but how do i explain that?

State an antiderivative $F:D\u27f6\mathbb{C}$ or explain why an antiderivative does not exist.

a) $f(z)={z}^{2},D=\mathbb{C}$

b) $f(z)=z\mathrm{sin}z,D=\mathbb{C}$

c) $f(z)=|z{|}^{2},D=\mathbb{C}$

d) $f(z)=\overline{z},D=\mathbb{C}$

For a) and b) I just performed simple integration for the equations

a) ${z}^{3}/3+C$ and b) $-z\mathrm{cos}z+\mathrm{sin}z+C$ but I am stuck on c) and d). It seems like an antiderivative does not exist but how do i explain that?

Cristal Roth

Beginner2022-05-01Added 13 answers

Step 1

If $F:D\to \mathbb{C}$ is complex differentiable at all points of D then it is a holomorphic function in D, and its derivative F′ is holomorphic in D as well. (That is a consequence of, e.g., Cauchy's differentiation formula, and implies that F is infinitely often complex differentiable.)

Step 2

You can verify (e.g. with the Cauchy-Riemann equations) that the functions f in (c) and (d) are not holomorphic, and therefore not the derivative of a holomorphic function F.

The restriction $D=\mathbb{C}$ is irrelevant in these cases, $f(z)=|z{|}^{2}$ and $f(z)=\overline{z}$ are not holomophic in any open subset $D\subset \mathbb{C}$

If $F:D\to \mathbb{C}$ is complex differentiable at all points of D then it is a holomorphic function in D, and its derivative F′ is holomorphic in D as well. (That is a consequence of, e.g., Cauchy's differentiation formula, and implies that F is infinitely often complex differentiable.)

Step 2

You can verify (e.g. with the Cauchy-Riemann equations) that the functions f in (c) and (d) are not holomorphic, and therefore not the derivative of a holomorphic function F.

The restriction $D=\mathbb{C}$ is irrelevant in these cases, $f(z)=|z{|}^{2}$ and $f(z)=\overline{z}$ are not holomophic in any open subset $D\subset \mathbb{C}$

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function

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