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mathematics
advanced engineering mathematics

Questions and Answers of Advanced Engineering Mathematics

Determine the location of the singularities, including those at infinity. For poles also state the order. Give reasons.(z - π)-1 sin z
Find and sketch or graph the image of the given region under the mapping w = cos z.-1 < x < 1, 0 ≤ y ≤ 1
Evaluate (counterclockwise). Show the details. z? sin z dz, dz. C the unit circle c 4z² – 1
Evaluate the following integrals and show details of your work. dx 2 - ix
Evaluate by the methods of this chapter. Show details. dx 1+ 4x* -0-
Verify Theorem 1 for f(z) = z-3 - z-1. Prove Theorem 1.
Find all points at which the mapping is not conformal. Give reason.exp (z5 - 80z)
Find and sketch the image of the region 2 ≤ |z| ≤ 3, π/4 ≤ θ ≤ π/2 under the mapping w = Ln z.
Evaluate (counterclockwise). Show the details. exp (-2?) -dz, C:|z| = 1.5 sin 4z
Evaluate by the methods of this chapter. Show details. dx x² – 4ix -00
Find the Cauchy principal value (showing details): dx x* + 3x2 .4
Find the magnification ratio M. Describe what it tells you about the mapping. Where is M = 1? Find the Jacobian J.w = 1/z2
(a) Integrating e-z2 around the boundary C of the rectangle with vertices -Î±, Î±, Î± + ib, -Î± + ib, letting Î± †’
Prove the statement in Prob. 27 for general k = 1, 2, · · ·. Use the Taylor series.Data from Prob. 27Let f(z) be analytic at z0. Suppose that f'(z0) = 0, · · ·, f(k-1)(z0) = 0. Then the
Find the Cauchy principal value (showing details): dx -00
Find the magnification ratio M. Describe what it tells you about the mapping. Where is M = 1? Find the Jacobian J.w = z +1/2z - 2
Find and sketch the image of the given region or curve under w = z2.1/√π < |z| < √π, 0 < arg z < π/2
Find and sketch the image of the given region or curve under w = z2.0 < y < 2
Find and sketch the image of the given region or curve under w = z2.y = -2, 2
Find and sketch the image of the given region or curve under w = 1/z.|z| < 1, 0 < arg z < π/2
Find and sketch the image of the given region or curve under w = 1/z.0 ≤ arg z ≤ π/4
Find and sketch the image of the given region or curve under w = 1/z.z = 1 + iy (-∞ < y < ∞)
Find the LFT that maps0, 2, 4 onto ∞, 1/2, 1/4, respectively
Find the LFT that maps-1, -i, i onto 1 - i, 2, 0, respectively
Find the fixed points of the mappingw = z4 + z - 64
Find the fixed points of the mappingw = (2iz - 1)/(z + 2i)
Find an analytic function w = f(z) that mapsThe quarter-disk |z| < 1, x > 0, y > 0 onto the exterior of the unit circle |w| = 1.
Find an analytic function w = f(z) that mapsThe interior of the unit circle |z| = 1 onto the exterior of the circle |w + 2| = 2.
Find an analytic function w = f(z) that mapsThe semi-disk |z| < 2, y > 0 onto the exterior of the circle |w - π| = π.
(a) Show that w = ˆšz has the values(b) Obtain from (18) the often more practical formulawhere sign y = 1 if y ‰¥ 0, sign y = -1 if y < 0, and all square roots of positive
Represent in polar form, with the principal argument.0.6 + 0.8i
Where does the power series converge uniformly? Give reason.
Is the given sequence z1, z2, · · ·, zn, · · · bounded?Convergent? Find its limit points. Show your work in detail..zn = (3 + 4i)n/n!
Where does the power series converge uniformly? Give reason.
Is the given sequence z1, z2, · · ·, zn, · · · bounded? Convergent? Find its limit points. Show your work in detail.zn = (1 + 2i)n
Where does the power series converge uniformly? Give reason. 00 E 2"(tanh n²)z2" 2n n-0
Find the center and the radius of convergence.
Is the given sequence z1, z2, · · ·, zn, · · · bounded? Convergent? Find its limit points. Show your work in detail.zn = (cos nπi)/n
Where does the power series converge uniformly? Give reason.
Find the center and the radius of convergence.
Is the given sequence z1, z2, · · ·, zn, · · · bounded?Convergent? Find its limit points. Show your work in detail.zn = [(1 + 3i)/√10]n
Prove that the series converges uniformly in the indicated region. 2n Σ 2n!' Izl s 1020 n-0
Find the center and the radius of convergence.
Is the given sequence z1, z2, · · ·, zn, · · · bounded? Convergent? Find its limit points. Show your work in detail.zn = sin (1/4nπ) + in
Prove that the series converges uniformly in the indicated region. 00 Σ Iz| s 1 n3 cosh n|z| 2-1
Find the radius of convergence. 4" -(г — пі)" п — 1 п-2 п
Find the center and the radius of convergence.
if z1, z2, · · · converges with the limit l and z*1, z*2 · · · converges with the limit show that z1 + z*1, z2 + z*2,· · · is convergent with the limit l + l*.
Find the radius of convergence in two ways:(a) Directly by the Cauchy€“Hadamard formula in Sec. 15.2.(b) From a series of simpler terms by using Theorem 3 or Theorem 4. 2n(2n – 1) 2n-2 Σ
Find the radius of convergence. Try to identify the sum of the series as a familiar function. 00 n-1
Find the center and the radius of convergence.
Is the given series convergent or divergent? Give a reason. Show details.
State clearly and explicitly where and how you are using Theorem 2.If f(z) in (2) is even (i.e., f(-z) = f(z)), show that αn = 0 for odd n. Give examples.
Find the Taylor series with center z0 and its radius of convergence.1/z, z0 = i
(a) Give a proof of Weierstrass M-test.(b) Derive Theorem 4 from Theorem 3.(c) Prove that uniform convergence of a series in a region G implies uniform convergence in any portion of G. Is the
Find the radius of convergence. Try to identify the sum of the series as a familiar function.
Find the center and the radius of convergence.
Is the given series convergent or divergent? Give a reason. Show details. η n-1 4.
State clearly and explicitly where and how you are using Theorem 2.Using (1 + z)p(1 + z)q = (1 + z)p+q, obtain the basic relation
Find the Taylor series with center z0 and its radius of convergence.cos2 z, z0 = π/2
(a) Formula (6) for R contains |αn/αn+1|, not |αn+1/αn|. How could you memorize this by using a qualitative argument?(b) What happens to R (0 < R < ∞) if you(i) Multiply all αn by k ≠
Show that (9) in Sec. 12.6 with coefficients (10) is a solution of the heat equation for t > 0 assuming that f(x) is continuous on the interval 0 ‰¤ x ‰¤ L and has
Is the given series convergent or divergent? Give a reason. Show details. 00 п+i Зп2 + 2i 31 п-0
State clearly and explicitly where and how you are using Theorem 2.(a) The Fibonacci numbers are recursively defined by α0 = α1 = 1, αn+1 = αn + αn-1 if n = 1, 2, · · ·.
Is the given series convergent or divergent? Give a reason. Show details.
Find the Taylor series with the given point as enter and its radius of convergence. z4, i
What is the difference between (7) and just stating |zn+1/zn| < 1?
Find the Taylor series with the given point as enter and its radius of convergence.1/z, 2i
Let |zn+1/zn| ‰¤ q < 1, so that the series z1+ z2+ · · · converges by the ratio test. Show that the remainder Rn= zn+1+ zn+2+ · · · satisfies
Determine the location and order of the zeros.(z4 - 81)3
Show that the Riemann surface of w = √(z - 1)(z - 2) has branch points at 1 and 2 sheets, which we may cut and join crosswise from 1 to 2. Introduce polar coordinates z - 1 = r1eiθ1 and z - 2 =
Show that substituting a linear fractional transformation (LFT) into an LFT gives an LFT.
Find the image of y = k = const, -∞ < x ≤ ∞, under w = ez.
Evaluate the following integrals and show the details of your work. de T + 3 cos
Determine the location and order of the zeros.tan2 2z
Find the branch points and the number of sheets of the Riemann surface.√iz - 2 + i
Find the image of x = k = const under w = 1/z. Use formulas similar to those in Example 1. y| y = 0 -21 -2 -1 -1, /1 12 T -1 -1 y= -2 x =0
Find and sketch the image of the given region under w = ez.0 < x < 1, 1/2 < y < 1
Evaluate the following integrals and show the details of your work. -27 1+ 4 cos 0 17 - 8 cos 0 do 0.
Obtain the mapping in Example 1 of this section from Prob. 18 in Problem Set 17.2.Data from Prob. 18Find all LFTs with fixed point(s).z = ±1
Determine the location and order of the zeros.cosh4 z
Find and sketch or graph the images of the given curves under the given mapping.x = 1, 2, 3, 4, y = 1, 2, 3, 4, w = z2
Find the branch points and the number of sheets of the Riemann surface.ln (6z - 2i)
Find the fixed points mentioned in the text before formula (5).
Find and sketch the image of the given region under w = ez.0 < x < ∞, 0 < y < π/2
Find all the singularities in the finite plane and the corresponding residues. Show the details.tan z
Evaluate the following integrals and show the details of your work. sin? e de 5 - 4 cos e 0,
Derive the mapping in Example 4 from (2). Find its inverse and the fixed points.
Find and sketch or graph the images of the given curves under the given mapping.|z| = 1/3, 1/2, 1, 2, 3, Arg z = 0, ±π/4, ±π/2, ±3π/2
Find the branch points and the number of sheets of the Riemann surface.e√z, √ez
Find the inverse z = z(w). Check by solving z(w) for w.w = z - i/z + i
Expand the function in a Laurent series that converges for 0 < |Z| < R and determine the precise region of convergence. Show the details of your work.ez/z2 - z3
Find all the singularities in the finite plane and the corresponding residues. Show the details.π/(z2 - 1)2
Evaluate the following integrals and show the details of your work. op 8 – 2 sin 0
Find the LFT that maps the given three points onto the three given points in the respective order.0, 1, 2 onto 1, 1/2, 1/3
Determine the location and order of the zeros.(z2 - 8)3(exp (z2) - 1)
Find the branch points and the number of sheets of the Riemann surface.√(4 - z2)(1 - z2)
Find the inverse z = z(w). Check by solving z(w) for w. z - ti -글iz-1
Sketch or graph the images of the lines x = 0, ±π/6, ±π/3, ±π/2 under the mapping w = sin z.
Find the Laurent series that converges for 0 < |z - z0| < R and determine the precise region of convergence. Show details. – 3i Zo = 3 (z – 3)2 22 - 3i

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