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andriy [413]
2 years ago
5

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

throughout any neighborhood lying in D. Suggestion: Suppose that f(z) does have a constant value ωo throughout some neighborhood in D.
Mathematics
1 answer:
alexandr1967 [171]2 years ago
6 0

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.

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When two balanced dice are rolled, there are 36 possible outcomes. Find the probability that the first die is 3 or that doubles
olga2289 [7]

Step-by-step explanation:

The space comprising all possible outcomes (sample space) is represented by the 6×66×6 matrix formed by the possible pairs representing the outcome of dice:

From the above sample space, there are six events wherein the first die is 44 (all elements of the fourth row in the matrix). That is, n(F)=6n(F)=6. Moreover, there are six events wherein doubles are rolled (all diagonal elements in the matrix). That is, n(D)=6n(D)=6. The result (4,4)(4,4) is the intersection of both events. That is, n(D∩F)=1n(D∩F)=1.

Therefore, the possibility/probability that the first die is 4 or that doubles are rolled is computed as follows:

P(D∪F)=P(D)+P(F)−P(D∩F)=n(D)n(S)+n(F)n(S)−n(D∩F)n(S)=636+636−136=1136P(D∪F)=P(D)+P(F)−P(D∩F)=n(D)n(S)+n(F)n(S)−n(D∩F)n(S)=636+636−136=1136

7 0
2 years ago
Which expression can you simplify by combining like terms?
omeli [17]

Answer:

The correct option is A.

Step-by-step explanation:

If the degree and variables of two terms are same, then they are called like terms.

In option A.

The given expression is  

12b^2+4ab-3ba-6

Here, 4ab and -3ab are like terms because both have same variables a and b with degree 1.

By combining like terms we get

12b^2+(4ab-3ab)-6

12b^2+ab-6

In options B, C and D, there are no like terms, So only the expression 12b^2+4ab-3ba-6 we can simplify by combining like terms.

Therefore the correct option is A.

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3 years ago
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Taya2010 [7]
EF will be the same as HG
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2 years ago
Find the three arithmetic means in this sequence. 14, __, __, __, 26
Alina [70]
26 - 14 = 12
12 ÷ 4 = 3

26 - 3 = 23
23 - 3 = 20
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14, 17, 20, 23, 26

Answer: The three missing arithmetic means in this sequence are 17, 20, 23.



6 0
3 years ago
Read 2 more answers
Sharon paid $78 sales tax on a new camera. If the sales tax is 6.5%, what was the cost of the camera?
Irina18 [472]

Answer:

$1200

Step-by-step explanation:

$1200

Step-by-step explanation:

78 = 6.5%

This means that if you divide 78 by 6.5, you get the equivalent of 1% of the price:

78 ÷ 6.5 = 12

So 1% = 12

Now simply multiply this by 100 to get the full answer:

12 x 100 = 1200

So the camera cost $1200!

5 0
3 years ago
Read 2 more answers
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