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advanced engineering mathematics

Advanced Engineering Mathematics 10th edition Erwin Kreyszig - Solutions

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 n-1
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 converse true?(d) Find the precise region of convergence of the series in Example 2 with x replaced by a
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 ≠ 0(ii) Multiply all αn by kn ≠ 0(iii) Replace αn by 1/αn? Can you think of an
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 one-sided derivatives at all interior points of that interval. Proceed as follows.Show that
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, · · ·. Find the limit of the sequence (αn+1/αn).(b) Compute a list of α1, · · ·, α12. Show that
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 the inequality |Rn| ‰¤ |zn+1|/(1 - q). Using this, find how many terms suffice for
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 = r2eiθ2, so that w = √r1r2ei(θ1+ θ2)/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
Find all the singularities in the finite plane and the corresponding residues. Show the details.z4/z2 - iz + 2
Evaluate the following integrals and show details of your work. 00 dx (1 + x2,3 -0
Find the LFT that maps the given three points onto the three given points in the respective order.0, -i, i onto -1, 0, ∞
(a) Show that if f(z) has a zero of order n > 1 at z = z0, then f'(z) has a zero of order n - 1 at z0.(b) Poles and zeros. Prove Theorem 4.(c) Show that the points at which a nonconstant analytic function f(z) has a given value k are isolated.(d) If f1(z) and f2(z) are analytic in a domain D and
Sketch or graph the given region and its image under the given mapping.1 < |z| < 3, 0 < Arg z < π/2, w = z3
Find and sketch or graph the image of the given region under w = sin z.-π/4 < x < π/4, 0 < y < 1
Find the fixed points.w = z - 3i
Find the Laurent series that converges for 0 < |z - z0| < R and determine the precise region of convergence. Show details. 1 Zo ?(z – i)'
Evaluate the following integrals and show details of your work. 00 dx (x2 – 2x + 5)2 00
Integrate counterclockwise around C. Show the details.e2/z, C:|z - 1 - i| = 2
Find the LFT that maps the given three points onto the three given points in the respective order.0, 2i, -2i onto -1, 0, ∞
Determine the location of the singularities, including those at infinity. For poles also state the order. Give reasons. e-i + z -i (z - i)3
Sketch or graph the given region and its image under the given mapping.x ≥ 1, w = 1/z
Find and sketch or graph the image of the given region under w = sin z.0 < x < π/6, -∞ < y < ∞
Find the fixed points.w = αz + b
Find the Laurent series that converges for 0 < |z - z0| < R and determine the precise region of convergence. Show details. az Zo = b – b' z- b'
Evaluate (counterclockwise). Show the details. z - 23 - dz, C: |z – 2 – i 22 - 4z – 5 = 3.2
Evaluate the following integrals and show details of your work. 00 .2 + 1 dx x* + 1
Integrate counterclockwise around C. Show the details.5z3/z2 + 4, C:|z - i| = πi/2
Find the LFT that maps the given three points onto the three given points in the respective order.-1, 0, 1 onto 1, 1 + i, 1 + 2i
Find the Maclaurin series and its radius of convergence. exp (z) exp (-1²) dt t
If z(x, y) = 300 - x2 - 9y2 [meters] gives the elevation of a mountain at sea level, what is the direction of steepest ascent at P: (4, 1)?
Given a curve C: r(t), find a tangent vector r'(t), a unit tangent vector u'(t), and the tangent of C at P. Sketch curve and tangent.r(t) = [t, t2, t3], P: (1, 1, 1)
Find the area of the quadrangle Q whose vertices are the midpoints of the sides of the quadrangle P with vertices A: (2, 1, 0), B: (5, -1, 0), C: (8, 2, 0), and D: (4, 3, 0). Verify that Q is a parallelogram.
When is the moment of a force equal to zero?
Let a = [1, 1, 0], b = [3, 2, 1], and c = [1, 0, 2]. Find the angle between:Find the angles of the triangle with vertices A: (0, 0, 2), B: (3, 0, 2), and C: (1, 1, 1). Sketch the triangle.
Find the unit vector in the direction of the resultant in Prob. 24.Data from Prob. 24Find the resultant in terms of components and its magnitude.p = [-1, 2, -3], q = [1, 1, 1], u = [1, -2, 2]
Find the length and sketch the curve.r(t) = [4 cos t, 4 sin t, 5t] from (4, 0, 0) to (4, 0, 10π).
Find the plane through the points A: (1, 2, 1/4), B: (4, 2, -2), and C: (0, 8, 4).
Find the velocity, speed, and acceleration of the motion given by r(t) = [3 cos t, 3 sin t, 4t] (t = time) at the point P: (3/√2, 3/√2, π).
Let a = [1, 1, 0], b = [3, 2, 1], and c = [1, 0, 2]. Find the angle between:Find the distance of the point A: (1, 0, 2) from the plane P: 3x + y + z = 9. Make a sketch.
Find the length and sketch the curve.r(t) = [α cos3 t, α sin3 t], total length.
Let f = xy - yz, v = [2y, 2z, 4x + z], and w = [3z2, x2 - y2, y2]. Find grad f and f grad f at P: (2, 7, 0)
Orthogonality is particularly important, mainly because of orthogonal coordinates, such as Cartesian coordinates, whose natural basis consists of three orthogonal unit vectors.For what c are 3x + z = 5 and 8x - y + cz = 9 orthogonal?
If |p| = 6 and |q| = 4, what can you say about the magnitude and direction of the resultant? Can you think of an application to robotics?
(a) Evaluate ∫C f(z) dz by Theorem 1 and check the result by Theorem 2, where:(i) f(z) = z4 and C is the semicircle |z| = 2 from 2i to 2i in the right half-plane,(ii) f(z) = e2z and C is the shortest path from 0 to 1 + 2i.(b) Experiment with a family of paths with common endpoints, say,
Find the type, transform to normal form, and solve. Show your work in detail.uxx + 2uxy + 10uyy = 0
Find the potential in the interior of the sphere r = R = 1 if the interior is free of charges and the potential on the sphere isf (Φ) = cos Φ
Solve for u = u (x, y):uxx + 25u = 0
Find the temperature in Prob. 11 with L = Ï€, c = 1, andf (x) = cos 2xData from Prob. 11€œAdiabatic€ means no heat exchange with the neighborhood, because the bar is completely insulated, also at the ends. Physical Information: The heat flux at the ends is
(a) Verify that u(x, t) = v(x + ct) + w(x - ct) with any twice differentiable functions v and w satisfies (1).(b) Verify that u = 1/√x2 + y2 + z2 satisfies (6) and u = ln (x2 + y2) satisfies (3). Is u = 1/√x2 + y2 a solution of (3)? Of what Poisson equation?
A small drum should have a higher fundamental frequency than a large one, tension and density being the same. How does this follow from our formulas?
(a) Write a program for calculating the Am’s in Example 1 and extend the table to m = 15. Verify numerically that αm ≈ (m - 1/4)π and compute the error for m = 1, · · ·, 10.(b) Graph the initial deflection f(r) in Example 1 as well as the first three partial sums of the series. Comment on
Find the deflection u(x, y, t) of the square membrane of side π and c2 = 1 for initial velocity 0 and initial deflection0.01 sin x sin y
Find the electrostatic potential between coaxial cylinders of radii r1 = 2 cm and r2 = 4 cm kept at the potentials U1 = 220 V and U2 = 140 V, respectively.
Find the temperature w(x, t) in a semi-infinite laterally insulated bar extending from x = 0 along the x-axis to infinity, assuming that the initial temperature is 0, w(x, t) †’ 0 as x †’ ˆž for every fixed t ‰¥ 0, and w(0, t) = f(t). Proceed as follows.Show

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