Step by step explanation

Use the theorem in Sec. 28 to show that if f (z) is analytic and not constant throughout a domain D, then it cannot be constant

alexandr1967 [171]

Answer:

The value of f(z) is not constant in any neighbourhood of D. The proof is as explained in the explaination.

Step-by-step explanation:

Given

For any given function f(z), it is analytic and not constant throughout a domain D

To Prove

The function f(z) is non-constant constant in the neighbourhood lying in D.

Proof

1-Assume that the value of f(z) is analytic and has a constant throughout some neighbourhood in D which is ω₀

2-Now consider another function F₁(z) where

F₁(z)=f(z)-ω₀

3-As f(z) is analytic throughout D and F₁(z) is a difference of an analytic function and a constant so it is also an analytic function.

4-Assume that the value of F₁(z) is 0 throughout the domain D thus F₁(z)≡0 in domain D.

5-Replacing value of F₁(z) in the above gives:

F₁(z)≡0 in domain D

f(z)-ω₀≡0 in domain D

f(z)≡0+ω₀ in domain D

f(z)≡ω₀ in domain D

So this indicates that the value of f(z) for all values in domain D is a constant ω₀.

This contradicts with the initial given statement, where the value of f(z) is not constant thus the assumption is wrong and the value of f(z) is not constant in any neighbourhood of D.

6 0

2 years ago

A population of rabbits is described by the function R(t) = 100(2t/5), where t is measured in months and R is measured in rabbit

Savatey [412]

Answer and Step-by-step explanation: The graph is shown in the attachment.

a. ΔR on [1,2] is mathematically expressed as:

ΔR = R(2) - R(1)

which means difference of population of rabbits after 2 months and after 1 month.

$R(1) = 100(\frac{2}{5}.1 )$

$R(1) = 100(\frac{2}{5} )$

R(2) = $100(\frac{2}{5}.2 )$

$R(2) = 100(\frac{4}{5} )$

$\Delta R = 100(\frac{4}{5} )-100(\frac{2}{5} )$

$\Delta R = 100[\frac{4}{5} - \frac{2}{5} ]$

$\Delta R=$ 40

Difference of rabbits between first and second months is 40.

b. R(0) = 100( $\frac{2}{5} .0$ )

R(0) = 0

Initially, there no rabbits in the population.

c. R(10) = $100(\frac{2}{5}.10 )$

R(10) = 400

In 10 months, there will be 400 rabbits.

d. R(t) = 500

$500=100(\frac{2}{5}.t )$

$\frac{500}{100}=\frac{2}{5}.t$

$t = \frac{500.5}{100.2}$

t = 12.5

In 12 and half months, population of rabbits will be 500.

3 0

3 years ago

Tell whether x and y are proportional,

Eduardwww [97]

Yes, as the value of x increases so does the value of y hence they are (directly) proportional

8 0

3 years ago

Max measures the diagonal of a square room to be 32feet find the perimeter and area of the room round answers to the nearest ten

Rasek [7]

Hello!

To find the side length of a square with the diagonal you use the equation

$a = \sqrt{2} \frac{d}{2}$

a is side length
d is diagonal length

Put in the values you know

$a = \sqrt{2} * \frac{32}{2}$

Divide

$a = \sqrt{2} * 16$

Multiply

a = 22.6

All the sides are 22.6 feet

Add all the sides

22.6 + 22.6 + 22.6 + 22.6 = 90.4

The answer is 90.4

Hope this helps!

3 0

3 years ago

The parallelogram below is being covered with carpet. The carpet is priced at $2.80 per square meter. How much will it cost to c

Sloan [31]

Answer:

$79.38

Step-by-step explanation:

Find the diagram of the question attached

Area of the parallelogram = Base * Height

Given

Base = 8.1m

Height = 3.5m

Area of the parallelogram = 8.1 * 3.5

Area of the parallelogram = 28.35m²

If the carpet is priced at $2.80 per square meter, then;

1m² = $2.80

28.35m² = x

x = 28.35 * 2.80

x = $79.38

Hence the cost of the carpet is $79.38

7 0

2 years ago