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mathematics
advanced engineering mathematics

Questions and Answers of Advanced Engineering Mathematics

Formulas (4), (5), and (11) (below) are needed from time to time. Derive
Find f(z) = u(x, y) + iv(x, y) with u or v as given. Check by the Cauchy–Riemann equations for analyticity.u = exp(-(x2 - y2)/2) cos xy
Solve and graph the solutions. Show details.z2 + z + 1 - i = 0
Find the value of:cos (3 - i)
Solve and graph the solutions. Show details.z4 - 6iz2 + 16 = 0
Find the value of:tan i
Inequalities and equalityProve (6).
Find the value of:cosh (π + πi)
Inequalities and equalityProve and explain the name|z1 + z2|2 + |z1 - z2|2 = 2(|z1|2 + |z2|2).
Integrate counterclockwise around the unit circle.
Integrate z2/(z2 - 1) by Cauchy’s formula counterclockwise around the circle.|z + 1| = 1
Find the path and sketch it.z(t) = (1 + 1/2i)t (2 ≤ t ≤ 5)
What is a parametric representation of a curve? What is its advantage?
What did we assume about paths of integration z = z(t)? What is ż = dz/dt geometrically?
Integrate counterclockwise around the unit circle.
Integrate z2/(z2 - 1) by Cauchy’s formula counterclockwise around the circle.|z + i| = 1.4
Find the path and sketch it.z(t) = t + 2it2 (1 ≤ t ≤ 2)
Can we conclude from Example 4 that the integral is also zero over the contour in Prob. 1?Data from Prob. 1Verify Theorem 1 for the integral of over the boundary of the square with vertices ±1 ±i.
State the definition of a complex line integral from memory.
Can you remember the relationship between complex and real line integrals discussed in this chapter?
Integrate counterclockwise around the unit circle.
Integrate the given function around the unit circle.(cos 3z)/(6z)
Find the path and sketch it.z(t) = 3 - i + √10e-it (0 ≤ t ≤ 2π)
What is the connectedness of the domain in which (cos z2)/(z4 + 1) is analytic?
How can you evaluate a line integral of an analytic function? Of an arbitrary continous complex function?
What value do you get by counterclockwise integration of 1/z around the unit circle? You should remember this. It is basic.
Integrate counterclockwise around the unit circle.
Integrate the given function around the unit circle.z3/(2z - i)
Find the path and sketch it.z(t) = 2 + 4eπit/2 (0 ≤ t ≤ 2)
Can we conclude in Example 2 that the integral of 1/(z2 + 4) over(a) |z - 2| = 2(b) |z - 2| = 3 is zero?
Which theorem in this chapter do you regard as most important? State it precisely from memory.
What is independence of path? Its importance? State a basic theorem on independence of path in complex.
Integrate. Show the details. Begin by sketching the contour. Why?
Experiment to find out to what extent your CAS can do contour integration. For this, use(a) The second method in Sec. 14.1.(b) Cauchy’s integral formula.
Find the path and sketch it.z(t) = t + it3 (-2 ≤ t ≤ 2)
What is deformation of path? Give a typical example.
Integrate. Show the details. Begin by sketching the contour. Why?
Integrate f(z) counterclockwise around the unit circle. Indicate whether Cauchy’s integral theorem applies. Show the details.f(z) = exp (z2)
Don’t confuse Cauchy’s integral theorem (also known as Cauchy–Goursat theorem) and Cauchy’s integral formula. State both. How are they related?
Integrate counterclockwise or as indicated. Show the details.
Find a parametric representation and sketch the path.Segment from (-1, 1) to (1, 3)
What is a doubly connected domain? How can you extend Cauchy’s integral theorem to it?
Integrate f(z) counterclockwise around the unit circle. Indicate whether Cauchy’s integral theorem applies. Show the details.f(z) = 1/(2z - 1)
What do you know about derivatives of analytic functions?
Integrate f(z) counterclockwise around the unit circle. Indicate whether Cauchy’s integral theorem applies. Show the details.f(z) = z-3
Integrate. Show the details. Begin by sketching the contour. Why?
Integrate counterclockwise or as indicated. Show the details.
Find a parametric representation and sketch the path.Upper half of |z - 2 + i| = 2 from (4, -1) to (0, -1)
How did we use integral formulas for derivatives in evaluating integrals?
Integrate f(z) counterclockwise around the unit circle. Indicate whether Cauchy’s integral theorem applies. Show the details.f(z) = 1/(z4 - 1.1)
How does the situation for analytic functions differ with respect to derivatives from that in calculus?
Integrate f(z) counterclockwise around the unit circle. Indicate whether Cauchy’s integral theorem applies. Show the details.f(z) = 1/z̅
Integrate. Show the details. Begin by sketching the contour. Why?
Integrate counterclockwise or as indicated. Show the details.
Find a parametric representation and sketch the path.x2 - 4y2 = 4, the branch through (2, 0)
What is Liouville’s theorem? To what complex functions does it apply?
Integrate f(z) counterclockwise around the unit circle. Indicate whether Cauchy’s integral theorem applies. Show the details.f(z) = Im z
What is Morera’s theorem?
Integrate f(z) counterclockwise around the unit circle. Indicate whether Cauchy’s integral theorem applies. Show the details.f(z) = 1/(πz - 1)
Integrate. Show the details. Begin by sketching the contour. Why?
Integrate counterclockwise or as indicated. Show the details.
Find a parametric representation and sketch the path.|z + α + ib| = r, clockwise
If the integrals of a function f(z) over each of the two boundary circles of an annulus D taken in the same sense have different values, can f(z) be analytic everywhere in D? Give reason.
Integrate f(z) counterclockwise around the unit circle. Indicate whether Cauchy’s integral theorem applies. Show the details.f(z) = 1/|z|2
Integrate. Show the details. Begin by sketching the contour. Why?
Integrate counterclockwise or as indicated. Show the details.
Find a parametric representation and sketch the path.Parabola y = 1 - 1/4x2 (-2 ≤ x ≤ 2)
Integrate f(z) counterclockwise around the unit circle. Indicate whether Cauchy’s integral theorem applies. Show the details.f(z) = z3 cot z
Integrate by the first method or state why it does not apply and use the second method. Show the details.
Integrate by a suitable method.
Evaluate the integral. Does Cauchy’s theorem apply? Show details.
Integrate by the first method or state why it does not apply and use the second method. Show the details.
Integrate by a suitable method.
Evaluate the integral. Does Cauchy’s theorem apply? Show details. Use partial functions.
Integrate by the first method or state why it does not apply and use the second method. Show the details.
Integrate by a suitable method.
Evaluate the integral. Does Cauchy’s theorem apply? Show details.
Integrate by the first method or state why it does not apply and use the second method. Show the details.
Evaluate the integral. Does Cauchy’s theorem apply? Show details.
Integrate by the first method or state why it does not apply and use the second method. Show the details.
Integrate by a suitable method.
Integrate by the first method or state why it does not apply and use the second method. Show the details.
Integrate by a suitable method.
Evaluate the integral. Does Cauchy’s theorem apply? Show details.
Integrate by the first method or state why it does not apply and use the second method. Show the details.
Integrate by a suitable method.
Write programs for the two integration methods. Apply them to problems of your choice. Could you make them into a joint program that also decides which of the two methods to use in a given case?
Find an upper bound of the absolute value of the integral in Prob. 21. Data from Prob. 21 Integrate by the first method or state why it does not apply and use the second method. Show the details.
Are 1/z + z + z2 + · · · and z + z3/2 + z2 + z3 + · · ·power series? Explain
What is convergence test for series? State two tests from memory. Give examples.
Is the given sequence z1, z2, · · ·, zn, · · · bounded?Convergent? Find its limit points. Show your work in detail.zn = (1 + i)2n/2n
Which of the series in this section have you discussed in calculus? What is new?
Material in this section generalizes calculus. Give details.
(a) Fig. 368. Produce this exciting figure using your CAS. Add further curves, say, those of s256, s1024 ,etc. on the same screen. (b) Study the nonuniformity of convergence experimentally by
What is a power series? Why are these series very important in complex analysis?
Give all the details in the derivation of the series in those examples.
Write out the details of the proof on term wise addition and subtraction of power series.
Where does the power series converge uniformly? Give reason.
What are the only basically different possibilities for the convergence of a power series?
What is absolute convergence? Conditional convergence? Uniform convergence?

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