Use the Cauchy-Riemann equations to show that f(z)=bar(z) is not analytic.

Use the Cauchy-Riemann equations to show that f(z)=bar(z) is not analytic.

Question
Equations
asked 2021-02-26
Use the Cauchy-Riemann equations to show that \(\displaystyle{f{{\left({z}\right)}}}=\overline{{{z}}}\) is not analytic.

Answers (1)

2021-02-27
Step 1
A function is said to be analytic when Cauchy-Riemann equations are satisfied.
The Cauchy-Riemann equations are satisfied when \(\displaystyle{u}_{{x}}={v}_{{y}}{\quad\text{and}\quad}{v}_{{x}}=-{u}_{{y}}\).
The given function is \(\displaystyle{f{{\left({z}\right)}}}=\overline{{{z}}}\)
Rewrite the given function as follows.
\(\displaystyle{f{{\left({z}\right)}}}=\overline{{{z}}}\)
=x-iy
Here u(x,y)=x and v(x,y)=-y.
Step 2
Evaluate \(\displaystyle{u}_{{x}}\) as follows.
\(\displaystyle{u}_{{x}}=\frac{{\partial}}{{\partial{x}}}{\left({x}\right)}\)
=1
Thus, \(\displaystyle{u}_{{x}}={1}\).
Evaluate \(\displaystyle{u}_{{y}}\) as follows.
\(\displaystyle{u}_{{y}}=\frac{\partial}{{\partial{y}}}{\left({x}\right)}\)
=0
Thus, \(\displaystyle{u}_{{y}}={0}\).
Step 3
Evaluate \(\displaystyle{v}_{{x}}\) as follows.
\(\displaystyle{v}_{{x}}=\frac{\partial}{{\partial{x}}}{\left(-{y}\right)}\)
=0
Thus, \(\displaystyle{v}_{{x}}={0}\).
Evaluate \(\displaystyle{v}_{{y}}\) as follows.
\(\displaystyle{v}_{{y}}=\frac{\partial}{{\partial{y}}}{\left(-{y}\right)}\)
=-1
Thus, \(\displaystyle{v}_{{y}}=-{1}\).
Clearly, \(\displaystyle{u}_{{x}}\ne{v}_{{y}}\). So, the function did not satisfy Cauchy-Riemann equations.
Therefore, the given function is not analytic.
0

Relevant Questions

asked 2021-02-08
State Cauchy-Riemann equations. Show that f(z) x*+ iy' is not analytic anywhere but the Cauchy-Riemann equations are satisfied at the origin.
asked 2021-05-18
Verify Cauchy Riemann equations are satisfied function \(\displaystyle{\sin{{h}}}\) 4z
asked 2021-02-17
Determine whether the complex functions given below satisfy the Cauchy-Riemann equations (have derivatives). (?\(\displaystyle\in{C}\))
f(z)=z lm (z)
asked 2021-05-22
Determine which equations are linear equations in the variables x, y, and z. If any equation is not linear, explain why not.
\(3\cos x-4y+z=\sqrt{3}\)
asked 2021-03-29
Solve the system. If the system does not have one unique solution, also state whether the system is onconsistent or whether the equations are dependent.
2x-y+z=-3
x-3y=2
x+2y+z=-7
asked 2021-05-05

A random sample of \( n_1 = 14 \) winter days in Denver gave a sample mean pollution index \( x_1 = 43 \).
Previous studies show that \( \sigma_1 = 19 \).
For Englewood (a suburb of Denver), a random sample of \( n_2 = 12 \) winter days gave a sample mean pollution index of \( x_2 = 37 \).
Previous studies show that \( \sigma_2 = 13 \).
Assume the pollution index is normally distributed in both Englewood and Denver.
(a) State the null and alternate hypotheses.
\( H_0:\mu_1=\mu_2.\mu_1>\mu_2 \)
\( H_0:\mu_1<\mu_2.\mu_1=\mu_2 \)
\( H_0:\mu_1=\mu_2.\mu_1<\mu_2 \)
\( H_0:\mu_1=\mu_2.\mu_1\neq\mu_2 \)
(b) What sampling distribution will you use? What assumptions are you making? NKS The Student's t. We assume that both population distributions are approximately normal with known standard deviations.
The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.
(c) What is the value of the sample test statistic? Compute the corresponding z or t value as appropriate.
(Test the difference \( \mu_1 - \mu_2 \). Round your answer to two decimal places.) NKS (d) Find (or estimate) the P-value. (Round your answer to four decimal places.)
(e) Based on your answers in parts (i)−(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level \alpha?
At the \( \alpha = 0.01 \) level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
At the \( \alpha = 0.01 \) level, we reject the null hypothesis and conclude the data are statistically significant.
At the \( \alpha = 0.01 \) level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the \( \alpha = 0.01 \) level, we reject the null hypothesis and conclude the data are not statistically significant.
(f) Interpret your conclusion in the context of the application.
Reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver. (g) Find a 99% confidence interval for
\( \mu_1 - \mu_2 \).
(Round your answers to two decimal places.)
lower limit
upper limit
(h) Explain the meaning of the confidence interval in the context of the problem.
Because the interval contains only positive numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, we can not say that the mean population pollution index for Englewood is different than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains only negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is less than that of Denver.
asked 2021-05-19
The method of undetermined coefficients can sometimes be used to solve first-order ordinary differential equations. Use the method to solve the following equations.
\(y'-3y=5e^{3x}\)
asked 2021-04-06
Use Cramer’s Rule to solve (if possible) the system of linear equations.
13x-6y=17
26x-12y=8
asked 2021-03-10
Use Cramer's rule to solve the given system of linear equations.
\(\displaystyle{x}_{{{1}}}-{x}_{{{2}}}+{4}{x}_{{{3}}}=-{2}\)
\(\displaystyle-{8}{x}_{{{1}}}+{3}{x}_{{{2}}}+{x}_{{{3}}}={0}\)
\(\displaystyle{2}{x}_{{{1}}}-{x}_{{{2}}}+{x}_{{{3}}}={6}\)
asked 2021-03-29
What are the parametric equations for the intersection of the planes x-y-z=1 and 2x + 3y + z = 2 ?
...