(a) Find the Laurent series of the function 1/(z^2-9)(2+3) centered at z= -3. (b) Evaluate ∫C[-3,3] 1/(z^2-9)(2+3) dz.
Answers
The value of the contour integral is 2πi times 1/18, which simplifies to πi/9.By substituting z = -3 into the Laurent series, we find that the residue at z = -3 is 1/6.
(a) The Laurent series of the function 1/(z^2-9)(2+3) centered at z = -3 can be found by expanding the function as a power series. We start by factoring the denominator, which is (z+3)(z-3).
Since we are expanding the function centered at z = -3, we rewrite the denominator as (z-(-3))(z-3), which becomes (z+3)(z+3) after simplification.
Now, we can express the function as a partial fraction decomposition: 1/((z+3)(z+3)) = A/(z+3) + B/(z+3)^2.
To find the coefficients A and B, we can multiply both sides of the equation by (z+3)(z+3) and equate the numerators: 1 = A(z+3) + B.
Simplifying further, we have 1 = Az + 3A + B.
Comparing the coefficients of like powers of z, we find A = 1/6 and B = -1/9.
Therefore, the Laurent series of the function 1/(z^2-9)(2+3) centered at z = -3 is given by 1/6(z+3) - 1/9(z+3)^2.
(b) To evaluate the contour integral ∫C[-3,3] 1/(z^2-9)(2+3) dz, where C is the contour from -3 to 3, we can apply the residue theorem.
Since the integrand has simple poles at z = -3 and z = 3, we need to find the residues at these points.
The residue at z = -3 can be obtained by evaluating the limit of (z+3) times the function 1/(z^2-9)(2+3) as z approaches -3. By substituting z = -3 into the Laurent series, we find that the residue at z = -3 is 1/6.
Similarly, the residue at z = 3 is found to be -1/9.
According to the residue theorem, the contour integral ∫C[-3,3] 1/(z^2-9)(2+3) dz is equal to 2πi times the sum of the residues within the contour. In this case, the sum of the residues is 1/6 + (-1/9) = 1/18.
Therefore, the value of the contour integral is 2πi times 1/18, which simplifies to πi/9.
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Related Questions
ABCD is a cyclic quadrilateral. BC is produced so that AB=CE. If BD is the bisector of angle abc, prove that DBE is isosceles triangle
Answers
A proof to show that DBE is an isosceles triangle has been in the two-column proof below.
What is an isosceles triangle?In Mathematics and Geometry, an isosceles triangle simply refers to a type of triangle with two (2) sides that are equal in length and two (2) equal angles.
In this context, we can prove that triangle DBE is an isosceles triangle by completing the two-column proof with the following statements and reasons:
Statements Reasons______________________
∠ABD ≅ ∠DBC Given
AB = CE Given
∠BAD ≅ ∠DCE Exterior angle of cyclic quadrilaterals
is equal to the opposite interior angle
AD = CD Equal chords subtend equal angles
ΔABD ≅ ΔCDE SAS Postulate
BD = DE Corresponding sides of congruent triangles
ΔBDE is isosceles Two angles are congruent
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Missing information:
The question is incomplete and the complete question is shown in the attached picture.
Please help me with this,I will give you brainliest (only if you anwser first)
Answers
What is the area of the figure? The figure is not drawn to scale.
05.1 cm²
011.7 cm²
027.72 cm²
O55.44 cm²
Answers
Answer:
27.72 cm^2
Step-by-step explanation:
Parallelogram's area = base x height
= 3.3 x 8.4
= 27.72 cm^2
someone please help :)
Answers
Answer: 25
Step-by-step explanation:
they are corresponding angles
Find y when x = 2
y = 30 - 2x
Answers
y= 30 - 4
y = 26
What is the inverse of 16?
Answers
Additive inverse of 16 is -16.
Multiplicative Inverse of 16 is 1/16.
What is Inverse ?To produce a result equal to 0, add the original number and the additive inverse by simply reversing the sign of the integer. On the basis of negation of the original number, the characteristics of additive inverse are listed below. For instance, if x is the starting number, then -x is x's additive inverse.
A multiplicative inverse or reciprocal for a number x is a number that, when multiplied by x, produces the multiplicative identity, or 1. It is represented by 1/x or x/1. A fraction a/b has a multiplicative inverse of b/a. Divide 1 by the real number to obtain the multiplicative inverse of the number.
A function that "undoes" another function is called an inverse. In other words, if f(x) generates y, then y entered into the inverse of f creates x. An invertible function is one that has an inverse, and the inverse is indicated by the symbol \(f^{-1}\).
Additive inverse of 16 is -16.
Multiplicative Inverse of 16 is 1/16.
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jorge wants to warm up some tomato soup wich option is a better deal per ounce: one 30 ounce container soup for 5.28 or two 14 ounce container of soup for 2.38 each
Answers
To solve this we would divide the pirce of the soups by their container ounces
$2.38÷14=$.17 per ounce
$5.28÷30=$.176 per ounce
We do not use a thrid decmal place when talking about money so we would round $.176 (since the third decmial place is greater than or equal to 5) up to $.18
Therefore Two 14 ounce containers of soup is a better deal per ounce.
You can compare two marginal distributions to see if the corresponding two variables are related.T or F
Answers
You cannot compare two marginal distributions to see if the corresponding two are related . So, the statement is false .
The answer to the stated question is false . No , you cannot compare two marginal distributions to see if the corresponding two variables are related.
The given question is related to probability and statistics.
Coming to probability distribution , it is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events.
The marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variables in the subset without reference to the values of the other variables.
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A 6-foot diameter circle is divided
into six congruent sectors. What
is the area of each sector in square
inches? (Use 3.14 for 2.)
Answers
Answer:
the area is 1235
Step-by-step explanation:
Answer:
2714.33 sq in
explanation-
since circle is divided into 6 congruent sectors, area of each sector is 1/6th of area of circle.\(ar \: of \: sector \: = ar \: of \: circle \: \div 6 \\ = \pi \times 72 \times 72 \div 6 \\ = 2714.33\)Differential Equations - Diff
. The general solution of the equation A. sin(x+y)+x² + 2y² = c D. sin(x+y)+2x+4y=c cos(x+y)+2x+(cos(x+y)+ B. cos(x + y) + x² = c E. None 4y)y' = 0 is C. sin(x+y)+ y² = c
Answers
The general solution of the given differential equation is C. sin(x+y)+ y² = c.
The given differential equation is in the form of a first-order homogeneous linear differential equation. To solve it, we can separate the variables and integrate.
First, we rewrite the equation in a more suitable form by rearranging the terms:
sin(x+y) + y² = c
Next, we can separate the variables by moving all the terms involving y to one side and the terms involving x to the other side:
sin(x+y) = c - y²
To solve for y, we can take the arcsine of both sides:
x+y = arcsin(c - y²)
Now, we can isolate y by subtracting x from both sides:
y = arcsin(c - y²) - x
This equation represents the general solution of the given differential equation. It shows that the value of y depends on the value of x and the constant c. The equation involves the inverse sine function, indicating that the solution may consist of multiple branches.
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A (not necessarily convex) $11$-sided polygon can have at most how many right interior angles?
Be careful! Unlike the previous problem, the polygons here do not need to be convex.
Answers
The number of interior angles in any n-sided polygon will be equal to 'n'. In a 11-sided polygon there are total 11 interior angles out of which it can have at most 8 right angles.
What is a polygon?
Polygon is a closed two dimensional figure made up of straight line segments. The line segments of any polygon do not cross each other, otherwise it would be called as a complex polygon. The simple polygon is of two types namely-concave and convex. In concave atleast one angle is reflex(>180°) whereas in convex polygon all angles are less than 180°.
For any 'n' sided polygon there are 'n' number of sides and 'n' number of interior and exterior angles.
For 11 sided polygon there would be 11 sides and 11 angles.
To find the sum of all interior angles=(n-2)x 180
=(11-2) x 180
=1620°
Each angle of regular hendekagon(11-gon) would be=1620÷11
=147.27°
Now for any non-convex polygon, maximum number of right angles are calculated on the following basis:
For 11-sided concave polygon we need sum of all angles=1620°
If we assume 10 right angles it would make sum=10x90
=900
remaining one angle=1620-900
=720(not possible)
So it cant have 10 right angles.
Now, for 9 right angles sum=9x90
=810
remaining two angles sum=1620-810
=810
Each angle would get=405 (not possible >360)
Similarly for 8 right angles sum=720
Remaining 3 angles sum=1620-720
=900
each angle would be=900÷3
=300(possible for concave)
Hence we can say that 11-gon can have at most (n-3) or 8 right angles.
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A.22
B.183
C.246
D.213
Answers
Answer:
B. 183
Step-by-step explanation:
m<L = (1/2)[m(arc)KN - m(arc)KM)]
35 = (1/2)(177x - 107x)
70 = 70x
x = 1
m(arc)KN = 177x = 177
m(arc)NMK = 360 - m(arc)KN
m(arc)MNK = 360 - 177
m(arc)MNK = 183
from a collection of 50 store customers, 3 are to be chosen to receive a special gift. how many groups of 3 customers are possible?
Answers
The combination of number of ways of groups of 3 customers are 19600 .
What is permutation and combination?
Combination and permutation are two alternative strategies to divide up a collection of elements into subsets in mathematics. The components of the subset may be arranged in any order when combined.The subset's components are listed in a permutation in a certain order.The number of store customer is N = 50
The number of customer that receive a special gift is r = 3
The expression for the number of possible ways to choose 3 customers is
= ⁿCr
= ⁵⁰C₃
= 50 * 49 * 48/3 * 2 * 1
= 19600
Thus, the number of ways of groups of 3 customers are 19600.
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Help asap have lesss then 14 minutes please
Answers
Answer:
m∠B = 31°
Step-by-step explanation:
To find m∠B, we use the theory that in order to get it we must find the difference between the two arcs GM and VR and dividing it into 2
When putting this to equation we get:
m∠B = \(\frac{1}{2}\) * (VR - GM) = \(\frac{1}{2}\) * ( 94° - 32°) = \(\frac{1}{2}\)(62°) = 31°
according to smith travel research, the average hotel price in the united states in 2009 was $97.00. assume the population standard deviation is $18.00 and that a random sample of 40 hotels was selected. what is the probability that a randomly selected sample mean will be more than $100?
Answers
The probability that a randomly selected sample mean will be more than $100 = 0.14082.
We are given that according to a research institution, the average hotel price in a certain year was $97.00.
Assume the population standard deviation is $18.00 and that a random sample of 40 hotels was selected.
Here, Mean = μ = $97.00 and Standard deviation = σ = $18 and n = 40
(a) Standard error of the mean = σ/\(\sqrt{n}\) = 18/\(\sqrt{40}\) = 2.74
(b) Let X bar = Sample Mean
The z score probability distribution for sample mean is ;
Z = Xbar-μ/ σ/\(\sqrt{n}\) ~ N(0,1)
So, Probability that the sample mean will be less than $102 = P(X bar < $102)
P(X bar < 102) = P(Xbar-μ/ σ/\(\sqrt{n}\) < 102 - 97.00/ 18/\(\sqrt{40}\)) = P(Z < 0.74) = 0.7704
(c) Probability that the sample mean will be more than $104 =P(X bar > $104)
= P(X bar > 104) = P(Xbar-μ/ σ/\(\sqrt{n}\) < 104 - 97.00/ 18/\(\sqrt{40}\)) = P(Z > 1.47) = 1 - P(Z <= 1.47)
= 1 - 0.92922 = 0.0708
(d) Probability that the sample mean will be between $99 and $100 is given by = P($99 < X bar < $100) = P(X bar < $100) - P(X bar <= $99)
= P(X bar < 100) = P(Xbar-μ/ σ/\(\sqrt{n}\) < 100 - 97.00/ 18/\(\sqrt{40}\)) = P(Z < 0.01) = 0.50399
= P(X bar <= 99) = P(Xbar-μ/ σ/\(\sqrt{n}\) <= 99 - 97.00/ 18/\(\sqrt{40}\)) = P(Z <= -0.35) = 1 - P(Z <= 0.35)
= 1 - 0.63683 = 0.36317
Therefore, P($99 < X bar < $100) = 0.50399 - 0.36317 = 0.14082 .
Hence the answer is the probability that a randomly selected sample mean will be more than $100 = 0.14082.
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Magnitude statistics allow one to draw general conclusions about a population from information collected in a population sample.
Answers
inferential statistics allows for someone to draw conclusions about a population from the information collected in a population sample.
the qestion is incomplete .please read below to find the missing content
Which of the following allows for someone to draw conclusions about a population from the information collected in a population sample?
a. magnitude statistics
b. central tendency
c. inferential statistics
d. effect size
The population is the number of people living together in a place. The population of a city is the number of people living in that city. These people are called residents or residents. The population includes all individuals living in that particular area.
Population refers to the total number of organisms living in a particular area. Population helps us estimate the number of beings and know how to act accordingly. For example, knowing the exact population of a city allows us to estimate the number of resources required. Similarly, animals can do the same.
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What are the 5 steps in dividing rational expressions?.
Answers
The 5 steps in dividing rational expressions are explained below.
What are rational expressions?Rational expression seem to be parts that have factors in their denominators (and frequently numerators as well). For instance, x 2 x + 3 \dfrac{x^2}{x+3} x+3x2 start part, x, squared, separated by, x, furthermore, 3, end portion is a reasonable articulation.
Steps to divide the rational expression,1) Take the rational expression.
2) Rewrite the division as the product of the first rational expression then, the reciprocal of the second.
3) Factor the numerators and denominators of rational expression completely.
4) Multiply the numerators and denominators together.
5) Simplify by dividing out common factors.
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Write the equation for the line through the given point with the given slope.
(4, 0); m = 0
Answers
Answer: y = 0
Step-by-step explanation:
Find (g∘w)(x)and (w∘g)(x)for g(x)=7x−4 and w(x)=x2−7x+3(g∘w)(x)=(w∘g)(x)=
Answers
In order to calculate the composite function (gow)(x), that is, g(w(x)), we need to use w(x) as the input (value of x) in the function g(x).
So we have:
\(\begin{gathered} g(x)=7x-4 \\ g(w(x))=7w(x)-4 \\ g(w(x))=7(x^2-7x+3)-4 \\ g(w(x))=7x^2-49x+21-4 \\ g(w(x))=7x^2-49x+17 \end{gathered}\)Now, calculating (wog)(x), we have:
\(\begin{gathered} w(x)=x^2-7x+3 \\ w(g(x))=g(x)^2-7g(x)+3 \\ w(g(x))=(7x-4)^2-7\cdot(7x-4)+3 \\ w(g(x))=49x^2-56x+16-49x+28+3 \\ w(g(x))=49x^2-105x+47 \end{gathered}\)to find the quoteint 2/5 ÷ 1/4 you need to?
Answers
Answer:
2/5 ÷ 1/4 = 1/10
In circle L, shown below, PQ is a chord of the circle which measures 42 cm. What is the length of PL?
Round to the nearest tenth digit.
Include all of the following in your work for full credit.
(a) length of segment PM
(b) what circle property did you use to find the length of PM
(c) what formula/theorem did you use to calculate the length of segment PL
(d) all math used to calculate the length of segment PL
Answers
a. Length of segment PM=21cm and length of segment PL = 24.6cm.
What is segment?In geometry, a segment is a part of a line that is bounded by two distinct endpoints and contains all the points between them. It can also be defined as the portion of a line that connects two points.
According to given information:(a) Using the given information, PQ=42cm and we can see that M is midpoint of PQ. Therefore we can use midpoint formula
PQ=PM+MQ
42=2PM
PM=42/2
PM=21
(b) We used the property that the perpendicular bisector of a chord passes through the center of the circle, which means that LP is the perpendicular bisector of PQ.
(c) We used the Pythagorean theorem again to solve for PL.
\(PL^2 = PM^2 + LM^2\\\\PL^2 = 21^2 + 13^2\\\\PL^2 = 610\\\\PL = \sqrt{(610)\)
PL ≈ 24.6 cm
(d)
\(PM^2=PL^2-LM^2= 610-169=441\\\\PM=21\)
Using the fact that LP is the perpendicular bisector of PQ, we can split PQ in half to get two segments of length 21. Then, using the Pythagorean theorem in right triangle LPL', where L' is the midpoint of PQ, we can solve for PL.
PL ≈ 24.6 cm
Therefore, the length of PL is approximately 24.6 cm.
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Alana and two of her friends evenly split the cost of a meal. They determine that each person owes $14.68. How much was the entire cost of the meal
What is the equation?
Answers
Answer:
Step-by-step explanation:
Alana and 2 friends evenly split the meal. This means there are 3 people in total.
Since they said each person owes $14.68. To find the total you just multiply $14.68 by 3.
Total = $14.68 x 3 = $?
What is the slope of a line that goes through (3, -2) and (-2, 5) ?
Answers
Answer:
-7/5
Step-by-step explanation:
m=(y2-y1)/(x2-x1)
m=(5-(-2))/(-2-3)
m=(5+2)/-5
m=7/-5
7y - 2 3/4 = 1/2 solve these equation
Answers
Answer:
y = \(\frac{13}{28}\)
Step-by-step explanation:
Given
7y - 2 \(\frac{3}{4}\) = \(\frac{1}{2}\) ( change the mixed number to an improper fraction )
7y - \(\frac{11}{4}\) = \(\frac{1}{2}\) ( multiply through by 4 to clear the fractions )
28y - 11 = 2 ( add 11 to both sides )
28y = 13 ( divide both sides by 28 )
y = \(\frac{13}{28}\)
Answer: y= 13/28
1. Convert 2 3/4 to improper fraction. Use this rule: a b/c = ac+b/c.
7y - 2 • 4 + 3 / 4 = 1/2
2. Simplify 2 • 4 to 8.
7y - 8 + 3 / 4 = 1/2
3. Simplify 8 + 3 to 11.
7y - 11/4 = 1/2
4. Add 11/4 to both sides.
7y = 1/2 + 11/4
5. Simplify 1/2 + 11/4 to 13/4.
7y = 13/4
6. Divided both sides by 7.
Y= 13/4 / 7
7. Simplify 13/4 / 7 to 13 / 4 • 7.
Y = 13 / 4 • 7
8. Simplify 4 • 7 to 28.
Y= 13/28
What is the rate of increase in water pressure in atm/km
Answers
Answer:
Rate of increase in water pressure is 992
Step-by-step explanation:
1. David uses 1 tbsp of sugar for every 1.25 cups of water when he makes lemonade complete the table below by filling in the missing values
Answers
Answer:
Step-by-step explanation:
David uses 1 table spoon of sugar for every 1.25 cups of water to prepare lemonade.
Therefore, Number of table spoons of sugar required (S) ∝ Cups of water required (w)
⇒ S ∝ w
⇒ S = kw
⇒ k = \(\frac{S}{w}\)
⇒ k = \(\frac{1}{1.25}\) = 0.8
So the formula to calculate the amount of water will be,
S = 0.8w
⇒ w = \(\frac{S}{0.8}\)
where k = proportionality constant
Now we can complete the table with the help of the formula.
Sugar (tbsp) Water (cups)
1 1.25
7 8.75
3 3.75
5 6.25
Is 3 a solution to 3x + 5 = 5x + 6 - 2
Answers
Answer:
no
Step-by-step explanation:
it's X= 0.5
group like terms
3x - 5x = 6 - 2 - 5
simplify the equation
-2x = -1
divide through by the coefficient of X
x = 0.5
Given the following functions, find and simplify (f. g)(5.5).
Provide your answer below:
(f-g)(5.5)=
f(x) = -x + 6
g(x) = -12x - 6
Answers
Answer:
78
Step-by-step explanation:
To compute (f.g)(5.5) , first compute g(5.5) , then use that value in f(x) to compute the composite function value
g(5.5) = -12(5.5) - 6 = -66 - 6 = -72
Use this value for x in f(x) = -x + 6
f(g(5.5)) = f(-72) = -(-72) + 6 = 72 + 6 = 78
Factor the algebraic expression. 7x+63
Answers
Step-by-step explanation:
7x+63
= 7 (x +9)
Hope it helps ya
Answer:
7(x+9)
Step-by-step explanation:
7 times what = 63?
7*9 = 63 so lets pull out a 7 from the expression
7x+63 = 7(x+9)
Anna buys ice cream and oranges at the store. • She pays a total of $59. 53. She pays a total of $6. 49 for the ice cream. She buys 8 bags of oranges that each cost the same amount. . . Write and solve an equation which can be used to determine x, how much each bag of oranges costs.
Answers
Answer: $6.63
Step-by-step explanation:
Total= $59.53
Ice cream total= $6.49
Orange bags total= x ($53.04)
(finding the amount paid in total for oranges)
$59.53 - $6.49= $53.04
(finding how much each bag costs)
She buys 8 bags of oranges-
$53.04 divided by 8
$6.63 for each bag of oranges
(hope this helps :)