Suppose that f(z) is analytic and nonzero in a domain D. Prove that In |f(z)l is harmonic in D.
Answers
The statement is about the harmonic nature of the function In |f(z)| in a domain D, assuming that f(z) is analytic and nonzero in D.
To prove that In |f(z)| is harmonic in the domain D, we need to show that it satisfies the Laplace's equation. In other words, we need to show that its Laplacian is zero.
Let's consider the function g(z) = In |f(z)|, where f(z) is the given analytic and nonzero function in D.
The Laplacian of a function g(z) in two dimensions is defined as the sum of the second partial derivatives of g with respect to the real and imaginary components of z.
∇²g = (∂²g/∂x²) + (∂²g/∂y²)
To show that In |f(z)| is harmonic, we need to prove that its Laplacian (∇²g) is zero.
Taking the partial derivatives of g(z) with respect to x and y, we have:
∂g/∂x = ∂/∂x [In |f(z)|]
∂g/∂y = ∂/∂y [In |f(z)|]
Using the chain rule, we can write these derivatives as:
∂g/∂x = (∂/∂x) [In |f(z)|] = (∂/∂x) [In |f(x+iy)|] = (∂/∂x) [In √(f(x+iy) * f(x+iy))]
= (∂/∂x) [In √(f(x+iy) * f(x-iy))]
Similarly, we can find ∂g/∂y.
Now, we calculate the Laplacian (∇²g) by taking the second partial derivatives:
∇²g = (∂²g/∂x²) + (∂²g/∂y²)
Substituting the expressions for the partial derivatives, we get:
∇²g = (∂²g/∂x²) + (∂²g/∂y²) = (∂/∂x) [ (∂g/∂x) ] + (∂/∂y) [ (∂g/∂y) ]
By simplifying these expressions, we can see that the Laplacian (∇²g) is equal to zero.
Therefore, we can conclude that In |f(z)| is a harmonic function in the domain D, given that f(z) is analytic and nonzero in D.
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Related Questions
Ethan is watering his plants. He starts with a watering can filled with 32 ounces of water, and he pours 3.5 ounces onto each plant. When Ethan finishes watering his plants, he has 14.5 ounces of water left. How many plants does he have? Write and solve an equation to find the answer.
Answers
The number of plants Ethan has are 5.
What is Equation?Two or more expressions with an Equal sign is called as Equation.
Let x be the number of plants Ethan has.
Ethan uses 3.5 ounces of water for each plant, so he will use a total of 3.5p ounces of water.
If he starts with 32 ounces of water and ends up with 14.5 ounces
Total of 32 - 14.5 = 17.5 ounces of water.
As given the amount of water used is equal to the amount of water per plant times the number of plants:
3.5x = 17.5
Divide both sides by 3.5
x=5
Hence, , Ethan has 5 plants.
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complete the square to rewrite the following equation in standard form
Answers
By completing the square, the equation in standard form is (x - 2)² + (y + 4)² = 4².
What is the equation of a circle?In Mathematics and Geometry, the standard form of the equation of a circle is modeled by this mathematical equation;
(x - h)² + (y - k)² = r²
Where:
h and k represent the coordinates at the center of a circle.r represent the radius of a circle.From the information provided below, we have the following equation of a circle:
x² - 4x + y² + 8y = -4
x² - 4x + (-4/2)² + y² + 8y + (8/2)² = -4 + (-4/2)² + (8/2)²
x² - 4x + 4 + y² + 8y + 16 = -4 + 4 + 16
(x - 2)² + (y + 4)² = 16
(x - 2)² + (y + 4)² = 4²
Therefore, the center (h, k) is (2, -4) and the radius is equal to 4 units.
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Complete Question:
Complete the square to rewrite the following equation in standard form. x² - 4x + y² + 8y = -4.
2 – 2n = 3n -+- 17
pls help
Answers
Answer:
n=-3
Step-by-step explanation:
Let's solve your equation step-by-step.
2−2n=3n+17
Step 1: Simplify both sides of the equation.
2−2n=3n+17
2+−2n=3n+17
−2n+2=3n+17
Step 2: Subtract 3n from both sides.
−2n+2−3n=3n+17−3n
−5n+2=17
Step 3: Subtract 2 from both sides.
−5n+2−2=17−2
−5n=15
Step 4: Divide both sides by -5.
\(\frac{-5n}{-5}\)=\(\frac{15}{-5}\)
n=−3
Mrs. Jacobson is shopping for school supplies with her children. Isabelle selected 4 one-inch binders and 5 two-inch binders, which cost a total of $24. Bill selected 1 one-inch binder and 2 two-inch binders, which cost a total of $9. How much does each size of binder cost?
Answers
ANSWER is one- incher=1 while two inch binder=2.By using simultaneous equation.
Mr. A sold his land to Mr.B at a profit of 10%. Mr.B. sold it to Mr.C at a gain of 5%. Mr.C.paid N1240 more for the house than Mr. A paid. What did Mr. A paid.
Answers
Answer:
Mr. A initially paid approximately N8000 for the land.
Step-by-step explanation:
Step 1: Let's assume Mr. A initially purchased the land for a certain amount, which we'll call "x" in currency units.
Step 2: Mr. A sold the land to Mr. B at a profit of 10%. This means Mr. A sold the land for 110% of the amount he paid (1 + 10/100 = 1.10). Therefore, Mr. A received 1.10x currency units from Mr. B.
Step 3: Mr. B sold the land to Mr. C at a gain of 5%. This means Mr. B sold the land for 105% of the amount he paid (1 + 5/100 = 1.05). Therefore, Mr. B received 1.05 * (1.10x) currency units from Mr. C.
Step 4: According to the given information, Mr. C paid N1240 more for the land than Mr. A paid. This means the difference between what Mr. C paid and what Mr. A paid is N1240. So we have the equation: 1.05 * (1.10x) - x = N1240
Step 5: Simplifying the equation: 1.155x - x = N1240
Step 6: Solving for x: 0.155x = N1240
x = N1240 / 0.155
x ≈ N8000
Therefore, in conclusion, Mr. A initially paid approximately N8000 for the land.
The seventh grade class help an end of the year dance. Of the 300 seventh graders, 240
decided to attend. What percent of the seventh grade class attended the end of the year dance?
Answers
Answer:
80 percent
Step-by-step explanation:
Dvide 300 by 240 which gives you 80 percent the amount of students that atended.
Answer:
80%
Step-by-step explanation:
240/300=.8
.8=80%
Jenna works as a hairdresser and each week she saves $45 from her paycheck. She
would like to save at least $540 before she goes on vacation.
45x > 540
How many weeks does she need to work before she can go on vacation?
Answers
Jenna needs to work for a minimum of 12 weeks before she can go on vacation and save at least $540.
To determine the number of weeks Jenna needs to work before she can go on vacation, we can set up an inequality based on the given information.
Let's define:
x = the number of weeks Jenna needs to work.
We know that Jenna saves $45 from her paycheck each week. Therefore, the amount she saves after x weeks can be calculated as 45x.
Jenna wants to save at least $540 before going on vacation. So we can write the inequality:
45x ≥ 540
To find the value of x, we can divide both sides of the inequality by 45:
x ≥ 540/45
Simplifying the right side of the inequality:
x ≥ 12
The result tells us that Jenna needs to work for at least 12 weeks in order to save $540 or more.
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(1,y) distance from (2, 3) is 5.
Answers
We will use the formula of the distance between 2 points
\(d=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2}\)We have
(2,3)=(x1,y1)
we can assume
(1,y)=(x2,y2)
\(\begin{gathered} 5=\sqrt[]{(1-2)^2+(y-3_{})^2} \\ 5=\sqrt[]{1+(y-3)^2} \\ 5^2=1+(y-3)^2 \\ 5^2-1=(y-3)^2 \\ 24=(y-3)^2 \\ \sqrt[]{24}=y-3 \\ We\text{ have two solutions} \\ y=3-2\sqrt[]{6}=-1.89 \\ y=3+2\sqrt[]{6}=7.89 \end{gathered}\)the two points that we can obtain
(1,-1.89) distance from (2,3) is 5
(1,7.89) distance from (2,3) is 5
A gallon of milk costs $3.32 and a container of orange juice costs $2.95. Max buys two gallons of milk and two containers of orange juice. How much does Max spend?
Answers
Step-by-step explanation: two bottles of milk is: $6.64 and two bottles juice: $5.90 so adding thos results $12.54
I hope help you
The table below shows how the number of university classrooms cleaned in an evening varies with the number of janitors:
a. What is the marginal product of the second janitor?
b. What is the average product of four janitors?
c. Is the addition of the third janitor associated with increasing, diminishing, or negative marginal returns? Explain.
d. Is the addition of the fourth janitor associated with increasing, diminishing, or negative marginal returns? Explain.
Answers
a. The marginal product of the second janitor is 4. b. The average product of four janitors is 8. c. Output between having three and two janitors d. output between having four and three janitors is -3.
The table provided shows the relationship between the number of janitors and the number of university classrooms cleaned in an evening. To answer the questions:
a. The marginal product of the second janitor can be found by calculating the difference in output between having two and one janitors. This is 12-8=4, so the marginal product of the second janitor is 4.
b. The average product of four janitors can be found by dividing the total output (32) by the number of janitors (4). Therefore, the average product of four janitors is 8.
c. The addition of the third janitor is associated with diminishing marginal returns. This can be seen by calculating the difference in output between having three and two janitors, which is 10-12=-2. The negative result indicates that the marginal product of the third janitor is lower than the previous one.
d. The addition of the fourth janitor is associated with negative marginal returns. This can be seen by calculating the difference in output between having four and three janitors, which is 7-10=-3. The negative result indicates that the marginal product of the fourth janitor is even lower than the previous one. This can be caused by a lack of space or equipment, or the fact that the workload cannot be divided equally among all the janitors.
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Line segment overline EG has a midpoint , F, at (- 3, 2) . The endpoint E is represented by the ordered pair (4, - 8) . What is the ordered pair that represents the endpoint G?
Answers
HELP ME PLZZZZZ ASAP
Answers
Answer:
Irrational
Step-by-step explanation:
What type of prism is this?
Answers
Answer: 3-dimensional prism
Step-by-step explanation:
Answer:
It is a 3D Triangular PrismRewrite the expression in the form z^n.
z^3/4 x z^2
Answers
Step-by-step explanation:
here's the answer to your question
HELP HELP PLS The population city in 2005 was 20,000. By 2015, the city's population had grown to
33,500. If the population growth follows a linear model, what is the projected
population for 2020?
Enter you answer as an integer
Answers
Answer:
40250
Step-by-step explanation:
Every 10 years the population goes up by 13,500. So divided 13500/2=6750. Then add6750 to 33500
Find the midpoint between points F(-5, -18) and K(15,8).
Answers
Answer:
(5,-5)
Step-by-step explanation:
(-5)+15= 10
10/2=5
(-18)+8=(-10)
-10/2=5
so your points would be 5,-5. you would add your x values and divide it by two and then your y values and divide them by two
Suppose that 1000 customers are surveyed and 850 are satisfied or very satisfied with a corporations products and services. Test the hypothesis H0: p = 0.9 against 1H: p not equals to 0.9 at alpha = 0.056. Find the P-value. Give your answers. null hypothesis The P-value is less than (choose the least possible).
Answers
The P-value for the hypothesis test H₀: p = 0.9 against H₁: p ≠ 0.9, at α = 0.056, is less than 0.056.
What is hypothesis test?
A hypothesis test is a statistical procedure used to make inferences or draw conclusions about a population based on sample data. It involves formulating two competing hypotheses, the null hypothesis (H₀)
In this hypothesis test, we are interested in determining if the proportion (p) of customers who are satisfied or very satisfied with a corporation's products and services is different from 0.9.
The null hypothesis (H₀) states that the proportion is equal to 0.9, while the alternative hypothesis (H₁) states that the proportion is not equal to 0.9.
To test these hypotheses, we use a binomial proportion test. From the given information, we have a sample of 1000 customers, and 850 of them are satisfied or very satisfied.
Using the sample proportion calculated as 850/1000 = 0.85, we can calculate the test statistic, which follows an approximately standard normal distribution under the null hypothesis.
Since the P-value is less than 0.056, we reject the null hypothesis and conclude that there is evidence to suggest that the proportion of satisfied or very satisfied customers is different from 0.9.
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Find the area of the complex figure.
Answers
Answer:
it's 189 the area
Step-by-step explanation:
area =189
Someone help solve please
Answers
Answer:
x = 9
Step-by-step explanation:
To solve x we use tan(60)
tan(60) = oppo/adj
tan(60) × adj = oppo
adj = oppo / tan(60)
x = 9√3 /tan(60)
x = 9
Answer:
x = 9y = 18Step-by-step explanation:
You want the shortest and longest sides of a 30°-60°-90° triangle that has a middle-length side that is 9√3 units.
Special right triangleThe triangle shown is one of two "special" right triangles. The sides have the ratios ...
1 : √3 : 2
If you multiply this set of ratios by 9, you get ...
9 : 9√3 : 18 = x : 9√3 : y
This tells you ...
x = 9
y = 18
__
Additional comment
The other "special" right triangle is the isosceles right triangle: 45°-45°-90°. Its side lengths have the ratios 1 : 1 : √2.
These triangles are useful to remember. They can help you remember the "short table of trig functions" (attached).
If you want to solve this using trig functions, you can use ...
Tan = Opposite/Adjacent
tan(60°) = 9√3/x
x = (9√3)/tan(60°) = (9√3)/(√3) = 9
and
Sin = Opposite/Hypotenuse
sin(60°) = 9√3/y
y = (9√3)/sin(60°) = (9√3)/(√3/2) = 18
What is the equation of the line that passes through the point (-1,6) and has a y-intercept of -5
Answers
well, since the y-intercept is at -5, or namely when the line hits the y-axis is at -5, that's when x = 0, so the point is really (0 , -5), and we also know another point on the line, that is (-1 ,6), to get the equation of any straight line, we simply need two points off of it, so let's use those two
\(\stackrel{y-intercept}{(\stackrel{x_1}{0}~,~\stackrel{y_1}{-5})}\qquad (\stackrel{x_2}{-1}~,~\stackrel{y_2}{6}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{6}-\stackrel{y1}{(-5)}}}{\underset{run} {\underset{x_2}{-1}-\underset{x_1}{0}}} \implies \cfrac{6 +5}{-1} \implies \cfrac{ 11 }{ -1 } \implies - 11\)
\(\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{(-5)}=\stackrel{m}{- 11}(x-\stackrel{x_1}{0}) \implies y +5 = - 11 ( x -0) \\\\\\ y+5=-11x\implies {\Large \begin{array}{llll} y=-11x-5 \end{array}}\)
please solve the answer
Answers
Answer:
x= -80
Step-by-step explanation:
Help!
Let f(x)= 7x² + 4x and g(x) = -9x -- 2. What function represents a new function, h(x), for which h(x)=(f+g)(x)
a. h(x)= -2x² + 2x
b. h(x) = 7x2 + 5x - 2
c. h(r)= -2x² + 4x - 2
d. h(x) = 7x- 5x - 2
Answers
Answer:
Option D
Step-by-step explanation:
h(x) = (f+g)(x) = f(x) + g(x)
= 7x² + 4x +(-9x-2)
= 7x² -5x -2
Rodrigo entrena para una maraton todos los dias sale a correr a una pista de 6 kilimetros de longitud,el lunes da dos vueltas,el martes da una vuelta y un medio,el miercoles da una vuelta y un tercio,el jueves da dos vueltas y dos quintos,el viernes da una vuelta y dos tercios,y el sabado dio dos vueltas y tres cuartos cuantos kilometros recorrio por dia
Answers
Answer:
11.65 km / day
Step-by-step explanation:
We can calculate the amount of daily kilometer in the following way
First day: 2 * 6 = 12 km
Second day: 1.5 * 6 = 9 km
Third day: 1.333 * 6 = 8 km
Fourth day: 2.4 * 6 = 14.4 km
Sixth day: 1.666 * 6 = 10 km
on Saturday: 2.75 * 6 = 16.5 km
the average per day
(12 + 9 + 8 + 14.4 + 10 +16.5) / 6 = 11.65
11.65 km/day
4/5 × 6/7 =
Indo mana indo? woy indo!
Answers
Answer:
6/7
Step-by-step explanation:
4/5 × 6/7
4 × 6/5 × 7
30/35
= 6/7 ☑
Answer:
The solution is 24/35
Step-by-step explanation:
\( \sf \frac{4}{5} \times \frac{6}{7} \\ = \sf \frac{4 \times 6}{5 \times 7} \\ = \sf \frac{24}{35} \)
By the definition of a parallelogram, ab∥dc. ad is a transversal between these sides, so ∠a and ∠d are angles. because ab and dc are , the same-side interior angles must be by the same-side interior angles theorem. therefore, ∠a and ∠d are supplementary.
Answers
In a parallelogram, The statement "angle A and angle D are supplementary" is not correct.
The same-side interior angles theorem states that if a transversal intersects two parallel lines, then the same-side interior angles are supplementary. However, in this case, we are dealing with a parallelogram, which has two pairs of parallel sides.
When we say that "ab||dc", we are stating that side AB is parallel to side DC. We cannot assume that side AD is a transversal between these two sides, as it could intersect the sides at some other angle. Therefore, we cannot conclude that angle A and angle D are angles.
However, we do know that opposite angles in a parallelogram are congruent. Therefore, we can conclude that angle A is congruent to angle C, and angle B is congruent to angle D.
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What are all the values that a correlation r can possibly take?A. 0 ≤ r ≤ 1B. r ≥0C. -1 ≤ r ≤ 1D. None of the above.
Answers
The values that a correlation r can possibly take are -1 ≤ r ≤ 1. Thus, option C is correct as per the correlation relation.
Correlation is a statistical measurement that describes the extent to which two variables are linearly related to each other in positive and negative directions. The correlation coefficient is denoted by r.
The value of correlation r can range from -1 to 1. Here,
-1 indicates: a perfect negative correlation
0 indicates: no linear correlation
1 indicates: a perfect positive correlation
Therefore, we can conclude that the values that a correlation r can possibly take are between -1 ≤ r ≤ 1.
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40 points please help!!!!
Answers
use the 3rd icon to measure 30 and 60 degrees, Then connect them with a line and add a right angle.
Answer:
Not Possible
Step-by-step explanation:
It is not possible. While it does add up to 180, if one is 90 degrees, the rest must be 45 and 45, or your not getting anything.
Let me know if this helped !!
What are ratios that are equivalent to 2:8
Answers
1:4 6:24 11:44 16:64 7:28 3:12 4:16
HELP PLEASE-
What is the average rate of change of the function over the interval x = 0 to x = 5?
f(x)=42x+1
Enter your answer, as a simplified fraction, in the boxes.
Answers
Answer:
42
Step-by-step explanation:
1+42(5)-(1+42(0)/ 5-0
=42
What number would X be in 5 + (x - 2) = -8
Answers
Answer:
-11
Step-by-step explanation:
Answer:
the answer is -11 sir
Step-by-step explanation: Let's solve your equation step-by-step.
5+x−2=−8
Step 1: Simplify both sides of the equation.
5+x−2=−8
5+x+−2=−8
(x)+(5+−2)=−8(Combine Like Terms)
x+3=−8
x+3=−8
Step 2: Subtract 3 from both sides.
x+3−3=−8−3
x=−11