question 2 1. Prove that f (z) = sin z is analytic.2. Check whether the function f(2) = e5² satisfies the Cauchy-Reimann equationsor not.3. Check whether the function f(z) = 26 satisfies the Cauchy-Reimann equationsor not.4. Check whether u(x, y) = y³ – 3æ²y is harmonic or not. If u(x, y) is harmonic,then find its conjugate harmonic v(x, y).5. Verify whether the function f(2) = 3x + y + i(3y – x) is entire or not.

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Description: question 2 1. Prove that f (z) = sin z is analytic.2. Check whether the function f(2) = e5² satisfies the Cauchy-Reimann equationsor not.3. Check whether the function f(z) = 26 satisfies the Cauchy-Reimann equationsor not.4. Check whether u(x, y) = y

Answered: Check whether the function f(2) = e5²…

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Check whether the function f(2) = e5² satisfies the Cauchy-Reimann equations or not.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.5: Solution Of Cubic And Quartic Equations By Formulas (optional)

Problem 29E

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Concept explainers

Addition Rule of Probability

It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.

Expected Value

When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).

Probability Distributions

Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.

Basic Probability

The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.

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Question

question 2

1. Prove that f (z) = sin z is analytic.
2. Check whether the function f(2) = e5² satisfies the Cauchy-Reimann equations
or not.
3. Check whether the function f(z) = 26 satisfies the Cauchy-Reimann equations
or not.
4. Check whether u(x, y) = y³ – 3æ²y is harmonic or not. If u(x, y) is harmonic,
then find its conjugate harmonic v(x, y).
5. Verify whether the function f(2) = 3x + y + i(3y – x) is entire or not.

Expert Solution

Step 1

The given function is $f (z) = e^{5 z}$ .

Check whether the function $f (z) = e^{5 z}$ satisfies the Cauchy-Riemann equations.

Step 2

Express the function $f (z) = e^{5 z}$ in the form $f (z) = u + i v$ as follows.

$\begin{array}{rcl} f (z) & = & e^{5 z} \\ = & e^{5 (x + i y)} \\ = & e^{5 x} \cdot e^{i 5 y} \\ = & e^{5 x} (\cos (5 y) + i \sin (5 y)) \\ = & e^{5 x} \cos (5 y) + i e^{5 x} \sin (5 y) \end{array}$

Thus, $\begin{array}{rcl} f (z) & = & e^{5 x} \cos (5 y) + i e^{5 x} \sin (5 y) \end{array}$ is of the form $f (z) = u + i v$ , where $u = e^{5 x} \cos (5 y) and v = e^{5 x} \sin (5 y)$ .

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