Carry out the following integrals, counterclockwise, around the indicated contour
Answers
For the first integral, z = π/4 is a pole of order 3 and lies inside the contour |z| = 1. Compute the residue:
[tex]\displaystyle \mathrm{Res}\left(\frac{e^z\cos(z)}{\left(z-\frac\pi4\right)^3}, z=\frac\pi4\right) = \lim_{z\to\frac\pi4}\frac1{(3-1)!} \frac{d^{3-1}}{dz^{3-1}}\left[e^z\cos(z)\right][/tex]
We have
[tex]\dfrac{d^2}{dz^2}[e^z\cos(z)] = -2e^z \sin(z)[/tex]
and so
[tex]\displaystyle \mathrm{Res}\left(\frac{e^z\cos(z)}{\left(z-\frac\pi4\right)^3}, z=\frac\pi4\right) = - \lim_{z\to\frac\pi4} e^z \sin(z) = -\frac{e^{\pi/4}}{\sqrt2}[/tex]
Then by the residue theorem,
[tex]\displaystyle \int_C \frac{e^z\cos(z)}{\left(z-\frac\pi4\right)^3} \, dz = 2\pi j \left(-\frac{e^{\pi/4}}{\sqrt2}\right) = \boxed{-\sqrt2\,\pi e^{\pi/4} j}[/tex]
For the second integral, z = 2j and z = j/2 are both poles of order 2. The second poles lies inside the rectangle, so just compute the residue there as usual:
[tex]\displaystyle \mathrm{Res}\left(\frac{\cosh(2z)}{(z-2j)^2\left(z-\frac j2\right)^2}, z=\frac j2\right) = \lim_{z\to\frac j2}\frac1{(2-1)!} \frac{d^{2-1}}{dz^{2-1}}\left[\frac{\cosh(2z)}{(z-2j)^2}\right] = \frac{16\cos(1)-24\sin(1)}{27}j[/tex]
The other pole lies on the rectangle itself, and I'm not so sure how to handle it... You may be able to deform the contour and consider a principal value integral around the pole at z = 2j. The details elude me at the moment, however.
Related Questions
From a sample size of 169, it was determined that the 68% confidence interval for the population mean is (505, 517). Therefore, the sample mean was ___. and the margin of error was ___.
A. 511; 6
B. 511; 12
C. 513; 6
D. 513; 12
Answers
Answer:
A. 511; 6
Step-by-step explanation:
The margin of error would be [tex]\pm\frac{517-505}{2}=\pm\frac{12}{2}=\pm 6[/tex]
The sample mean would be [tex]505+6=511[/tex] (or [tex]517-6=511[/tex])
In a group of workers, 16 belonged to a lottery syndicate and 4 did not.
What fraction of the workers did not belong to the syndicate?
1: 4/16
2: 1/5
3: 1/4
4: 3/4
Answers
Answer:
HI THEREthe answer is number 1 !!
• correct me if I'm wrong
Hi need help for this maths question
Answers
a) If f(y) is a probability density function, then both f(y) ≥ 0 for all y in its support, and the integral of f(y) over its entire support should be 1. eˣ > 0 for all real x, so the first condition is met. We have
[tex]\displaystyle \int_{-\infty}^\infty f(y) \, dy = \frac14 \int_0^\infty e^{-\frac y4} \, dy = -\left(\lim_{y\to\infty}e^{-\frac y4} - e^0\right) = \boxed{1}[/tex]
so both conditions are met and f(y) is indeed a PDF.
b) The probability P(Y > 4) is given by the integral,
[tex]\displaystyle \int_{-\infty}^4 f(y) \, dy = \frac14 \int_0^4 e^{-\frac y4} \, dy = -\left(e^{-1} - e^0\right) = \frac{e - 1}{e} \approx \boxed{0.632}[/tex]
c) The mean is given by the integral,
[tex]\displaystyle \int_{-\infty}^\infty y f(y) \, dy = \frac14 \int_0^\infty y e^{-\frac y4} \, dy[/tex]
Integrate by parts, with
[tex]u = y \implies du = dy[/tex]
[tex]dv = e^{-\frac y4} \, dy \implies v = -4 e^{-\frac y4}[/tex]
Then
[tex]\displaystyle \int_{-\infty}^\infty y f(y) \, dy = \frac14 \left(\left(\lim_{y\to\infty}\left(-4y e^{-\frac y4}\right) - \left(-4\cdot0\cdot e^0\right)\right) + 4 \int_0^\infty e^{-\frac y4} \, dy\right)[/tex]
[tex]\displaystyle \cdots = \int_0^\infty e^{-\frac y4} \, dy[/tex]
[tex]\displaystyle \cdots = -4 \left(\lim_{y\to\infty} e^{-\frac y4} - e^0\right) = \boxed{4}[/tex]
Q10. Five t-shirts and a hat cost £83.00. Two t-shirts and a hat cost £38.00. How much does one t-shirt cost?
No spam links and plz write down the answer.
Answers
Step-by-step explanation:
Two t-shirts that are(31.80) plus a hat(6.20) cost £38.00.
which means that one t-shirt costs £15.90 :)
will mark brainliest Select the correct answer.
Which relation is a function?
ОА
ху
14 -2
5 -3
6 -2
7-3
ОВ.
ху
310
220
3 30
440
ОС
Х У
-52
-53
-104
-105
OD.
ху
1015
2025
8 30
2035
Answers
Ilana has 70¢ in her purse. All the coins have the same value. What could describe all the coins she has in her purse?
71 pennies
8 dimes
14 nickels
3 quarters
Answers
Answer:
14 nickels is the answer
Step-by-step explanation:
How do you make a mixed fraction and write it out as a improper fraction?
Answers
Answer:
1) Multiply the whole number part by the fraction's denominator.
2) Add that to the numerator.
3) Then write the result on top of the denominator.
Step-by-step explanation:
Hope this helps!
Answer:
Explanation
|
Step-by-step explanation:
you you multiply the whole number by the denominator and then add the numerator to that number.
Solve the equation. Show your work!
2. 2m + 1.5 = 3m - 11.7
Answers
Answer:
gek
Step-by-step explanation:
a coffee mug collection includes four red mugs, eight yellow mugs and two white mugs. What is the probability of selecting two white mugs one after another?
Answers
Assuming that the selection process was done without replacement, the probability of selecting two white mugs without replacement is 1/28
Given dataSample Space
Red mugs = 4
yellow mugs = 8
white mugs = 2
Sample Size = 4+8+2 = 14
We are going to assume the selection was done with out replacement
Hence the probability of selecting the first white mug is
Pr(W1) = 2/8 = 1/4
The probability of selecting the second white mug is
Pr(W2) = 1/7
Hence the probability of selecting two white mugs one after another
= 1/4*1/7
=1/28
Learn more about probability at:
https://brainly.com/question/25839839
how can 26n-7m+4(10n-6m)
Answers
Answer:
66n-31m
Step-by-step explanation:
26n-7m+4(10n-6m)
26n-7m+40n-24m
66n-31m
When a number is subtracted from -3 2/3 , the result is -2 1/6. What is the number?
A.5 5/6
B. 1 1/2
C. - 1 !/2
D. -5 5/6
Answers
Answer:
D. [tex]-5 \frac{5}{6}[/tex]
Step-by-step explanation:
Add the values:
[tex]-3\frac{2}{3}+\left(-2\right)\frac{1}{6}[/tex]
[tex]=-\frac{11}{3}+\left(-2\right)\frac{1}{6}[/tex]
[tex]=-\frac{11}{3}-\frac{1}{3}[/tex]
[tex]-5 \frac{5}{6}[/tex]
Answer:
C. - 1 ½
Step-by-step explanation:
-3⅔ - x = - 2⅙
x = -3⅔ + 2⅙
x = -11/3 + 13/6
x = -22/6 + 13/6
x = -9/6
x = -1 ½
Please confirm my answer to be correct
Answers
Answer:
Actually, Corresponding angles are congruent.
Step-by-step explanation:
They are only supplementary in the special case where they are both 90°
Answer:
Option D
Step-by-step explanation:
Since we know that measure of corresponding angles are equal. So, corresponding angle are congruent to each other
Find the perimeter of the figure. Use 3.14 for 7 and round to at least 1 decimal place.
Answers
Answer:
the perimeter of the figure is 27.4 mi
Step-by-step explanation:
rectangle perimeter = 6+6+6 = 18
half circle perimeter = 1/2 2πr = 3.14 × 3 = 9.42
the perimeter of the figure is 18+9.42 = 27.42 --> 27.4 mi
in simplest terms: (-5x-7)-(-4x-6)(−5x−7)−(−4x−6)
Answers
Answer:
-20x^2-59x-43
Step-by-step explanation:
The gas tank on a car holds 13.3 gallons. If the car goes 199.5 miles on a single tank, how many miles per gallon does the car get?
Answers
Answer:
292.6
Step-by-step explanation:
How much money does drake make and how many houses does he have
Answers
Answer:
Step-by-step explanation:
Drake has 2 houses and makes $50,000,000
PLS HELP!!!
Express each of these numbers, as a product of two fractions
1/15
Answers
[tex]\dfrac 13 \times \dfrac 15 = \dfrac 1{15}[/tex]
The product of the two fraction numbers whose result is 1/15 will be 1/3 and 1/5.
What is Algebra?Algebra is the study of algebraic expressions, while logic is the manipulation of those concepts.
The acronym PEMDAS stands for Parenthesis, Exponent, Multiplication, Division, Addition, and Subtraction. This rule is used to answer the problem correctly and precisely.
The result of the product of the two fraction numbers is given below.
1/15
The factor of the number 15 is given as,
15 = 3 x 5
Then the expression is written as,
1/15 = 1/ (3 x 5)
1/15 = (1/3) x (1/5)
The multiplication of the two rational numbers whose result is 1/15 will be 1/3 and 1/5.
More about the Algebra link is given below.
https://brainly.com/question/953809
#SPJ2
At a school 60% of students buy a school lunch. Only 10% of students buy lunch and dessert. What is the probability that a student who buys lunch also buys dessert?
Answers
Answer:
1/6
Step-by-step explanation:
P(A) = probability a student buys lunch = 60% = 0.6
P(B) = probability a student buys lunch and dessert = 10% = 0.1
We are looking at the probability that a student buys lunch and dessert on the condition that they already bought lunch. Therefore, we are looking at a conditional probability.
P(B given A) = P(A and B) / P(A)
= 0.1 / 0.6
= 1/6 ≈ 0.167
How to determine if Triangles are Congruent?
If they are, how do you know if they are SSS or ASA or SAS or AAS?
Essentially, how do I apply the different congruent thermos to the questions attached?
Answers
So basically what you have to do is first identify all of the sides or angles that equal eachother then you can see if you did a side side and side in a row etc, like 44 would be SAS because the vertical angles would be inbetween the two sides. 46 would be SSS because after you make the line in the middle for reflexive property you can see that there are three sides in a row. If this didnt help i recommend looking on Khan Academy
A line that includes the point (0, -2) has a slope of 1. What is its equation in slope-intercept
form?
Write your answer using integers, proper fractions, and improper fractions in simplest form.
Answers
Answer:
y=1x-2
Step-by-step explanation:
Well this one is already given
When x=0, it's the y intercept
and it gives the slope, so we got it :D
Answer:
y=x-2
Step-by-step explanation:
y-y1=m(x-x1)
y-(-2)=1(x-0)
y+2=1(x)
y+2=x
y=x-2
I do not know how to do it
Answers
Answer:
this is do confusing I do not understand
What is the common ratio of the sequence?
-2, 6, -18, 54,...
A.-3
B.-2
C.3
D.8
Answers
A line includes the points (-7, 1) and (7, -1). What is its equation in slope-intercept form?
Write your answer using integers, proper fractions, and improper fractions in simplest form.
Answers
Answer:
y=-1/7x
Step-by-step explanation:
m=(y2-y1)/(x2-x1)
m=(-1-1)/(7-(-7))
m=-2/(7+7)
m=-2/14
simplify
m=-1/7
--------------------
y-y1=m(x-x1)
y-1=-1/7(x-(-7))
y-1=-1/7(x+7)
y=-1/7x-7/7+1
y=-1/7x-1+1
y=-1/7x
In the New York marathon there were 45 360 runners. 7 /10 of the runners were women. How many of the runners were men?
Answers
Answer:
13,608
Step-by-step explanation:
If 7/10 of the runners were women, than 3/10 of the runners were men. 3/10 of 45,360 runners is 13,608.
Hope this helps! Pls give brainliest! Also, plz sub to kgirl633 on yt.
basically how to write y-3=4(x+2) in slope intercept form
Answers
Answer: y = 4x + 11
Step-by-step explanation:
The slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept.
y = mx + b
Rewrite in slope-intercept form.
Simplify the right side.
Simplify 4(x + 2).
y - 3 = 4x + 8
Move all terms not containing y to the right side of the equation.
Add 3 to both sides of the equation.
y = 4x + 8 + 3
Add 8 and 3.
y = 4x + 11
Your Welcome! :)
What is an equation of the line that passes through the point (6,-4) and isperpendicular to the line 22 - y = 5?
Answers
i really need help, does anyone know these
Answers
Answer: If angle A is congruent to angle E
6(25-8w)+20w for w=2
Answers
150−28w
i think i dont know
what is the value of the expression when a=5, b=4, and c=6?
4a/2b-c
a. 2
b. 5
c. 9
d. 10
Answers
Answer:
D
Step-by-step explanation:
Pluck in the values, we have:
4.5/2.4 - 6 = 20/2 = 10
The second step to dividing 3 into 1/2
Answers
Answer:
Change the sign
Step-by-step explanation:
Asumming you are talking about the keep change flip method for dividing fraction, you would 1) keep 3/1, 2) flip the / symbol into * and 3) flip the 2nd fraction