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advanced engineering mathematics

Questions and Answers of Advanced Engineering Mathematics

Are the following functions analytic? Use (1) or (7).f(z) = 1/(z - z5)
Find, in the form u + iv,sin 2πi
Are the following functions analytic? Use (1) or (7).f(z) = Arg 2πz
Find Ln z when z equals1 ± i
Determine and sketch or graph the sets in the complex plane given by|z + i| ≥ |z - i|
Represent in polar form and graph in the complex plane as in Fig. 325. Do these problems very carefully because polar forms will be needed frequently. Show the details. -4 + 19i 2 + 5i У pl+i л/4
Find, in the form u + iv,cos πi, cosh πi
Let z1= -2 + 11i, z2= 2 - i. Showing the details of your work, find, in the form x + iy: Z122, (z1z2) Z1z2.
Are the following functions analytic? Use (1) or (7).f(z) = ln |z| + i Arg z
Write in exponential form (6):√i, √-i
Find Ln z when z equals-15 ± 0.1i
Find Re f, and Im f and their values at the given point z.f (z) = 5z2 - 12z + 3 + 2i at 4 - 3i
Determine the principal value of the argument and graph it as in Fig. 325.-5, -5 - i, -5 + i У pl+i л/4
Find, in the form u + iv,sinh (3 + 4i), cosh (3 + 4i)
Let z1 = -2 + 11i, z2 = 2 - i. Showing the details of your work, find, in the form x + iy:Re (1/z22 ), 1/Re (z22)
Are the following functions harmonic? If our answer is yes, find a corresponding analytic function f(z) = u(x, y) + iv(x, y).u = x2 + y2
Find all values and graph some of them in the complex plane.ln e
Find Re f, and Im f and their values at the given point z.f (z) = (z - 2)/(z + 2) at 8i
Find, in the form x + iy, showing details,(1 - i)10
Find, in the form u + iv,cos 1/2πi, cos [1/2π(1 + i)]
Let z1 = -2 + 11i, z2 = 2 - i. Showing the details of your work, find, in the form x + iy:z1/z2, z2/z1
Are the following functions harmonic? If our answer is yes, find a corresponding analytic function f(z) = u(x, y) + iv(x, y).v = xy
Find all values and graph some of them in the complex plane.ln (-7)
Find Re and Im ofe-πz
Find out, and give reason, whether f (z) is continuous at z = 0 if f(0) = 0 and for z ≠ 0 the function f is equal to:(Re z2)/|z|
Determine the principal value of the argument and graph it as in Fig. 325.-1 + 0.1i, -1 - 0.1i У pl+i л/4
Find, in the form x + iy, showing details,√i
Using the definitions, prove:|sinh y| ≤ |cos z| ≤ cosh y, |sinh y| ≤ |sin z| ≤ cosh y.Conclude that the complex cosine and sine are not bounded in the whole complex plane.
Let z1= -2 + 11i, z2= 2 - i. Showing the details of your work, find, in the form x + iy: Z1/22, (Z1/22)
Are the following functions harmonic? If our answer is yes, find a corresponding analytic function f(z) = u(x, y) + iv(x, y).u = sin x cosh y
Find all values and graph some of them in the complex plane.ln (4 + 3i)
Find out, and give reason, whether f (z) is continuous at z = 0 if f(0) = 0 and for z ≠ 0 the function f is equal to:(Im z2)/|z|2
Graph in the complex plane and represent in the form x + iy:6 (cos 1/3π + i sin 1/3π)
Find, in the form x + iy, showing details,eπi/2, e-πi/2
Find all solutions.sin z = 100
Are the following functions harmonic? If our answer is yes, find a corresponding analytic function f(z) = u(x, y) + iv(x, y).u = x3 - 3xy2
Solve for z.ln z = -πi/2
Re tan z and Im tan z. Show that
Find the value of the derivative of(iz3 + 3z2)3 at 2i
Determine α and b so that the given function is harmonic and find a harmonic conjugate.u = cos αx cosh 2y
Find the principal value. Show details.(2i)2i
Find all solutions and graph some of them in the complex plane. ez = -2
Find and graph all roots in the complex plane.3√3 + 4i
Find and graph all values of:√-32i
Determine α and b so that the given function is harmonic and find a harmonic conjugate.u = cosh αx cos y
Find the principal value. Show details.(1 - i)1+i
(a) Prove that (1) is equivalent to the pair of relations(b)(c)(d) If f(z) is differentiable at z0, show that f(z) is continuous at z0.(e) Show that f(z) = Re z = x is not differentiable at any z.
Find and graph all roots in the complex plane.4√-4
Find the principal value. Show details.(i)i/2
Find f(z) = u(x, y) + iv(x, y) with u or v as given. Check by the Cauchy–Riemann equations for analyticity.v = y/(x2 + y2)
Find the principal value. Show details.(3 + 4i)1/3
Find f(z) = u(x, y) + iv(x, y) with u or v as given. Check by the Cauchy–Riemann equations for analyticity.u = cos 3x cosh 3y
Solve and graph the solutions. Show details.z2 - (6 - 2i) z + 17 - 6i = 0
By definition, the inverse sine w = arcsin z is the relation such that sin w = z. The inverse cosine w = arccos z is the relation such that cow w = z. The inverse tangent, inverse cotangent, inverse
Let f(z) be analytic. Prove that each of the following conditions is sufficient for f(z) = const.(a) Re f(z) = const(b) Im f(z) = const(c) f'(z) = 0
Find f(z) = u(x, y) + iv(x, y) with u or v as given. Check by the Cauchy–Riemann equations for analyticity.v = cos 2x sinh 2y
Solve and graph the solutions. Show details.z4 + 324 = 0. Using the solutions, factor z4 + 324 into quadratic factors with real coefficients.
(a) Write f(z) in terms of partial fractions and integrate it counterclockwise over the unit circle, where(b) In the light of the principle of deformation of path. Then consider another family of
Gain additional insight into the proof of Cauchy€™s integral theorem by producing (2) with a contour enclosing z0(as in Fig. 356) and taking the limit as in the text. Chooseand (c) another
Integrate. Show the details. Begin by sketching the contour. Why? exp (z) dz, Jc z(z – 21)2 C: z - 3i| = 2 clockwise.
Integrate counterclockwise or as indicated. Show the details. C the circle with center dz, z2 + 4z + 3 radius 2 -1 and
Integrate by the first method or state why it does not apply and use the second method. Show the details. cos 2z dz, C the semicircle Izl = TT, x20 from
Integrate by a suitable method. Re z dz from 0 to 3 + 27i along y = x
Evaluate the integral. Does Cauchy€™s theorem apply? Show details.Use partial functions. dz C:
Integrate by the first method or state why it does not apply and use the second method. Show the details. (z + z-1) dz, C the unit circle, counterclockwise
Integrate by a suitable method. (2? + + z?) dz from z = 0 horizontally to z = 2, then vertically upward to 2 + 2i.
Evaluate the integral. Does Cauchy€™s theorem apply? Show details. coth z dz, C the circle |z – mi| = 1 clockwise. Jc
Evaluate the integral. Does Cauchy€™s theorem apply? Show details. tan z -dz, C the boundary of the square with 4 - 16 vertices +1, ±i clockwise.
Evaluate the integral. Does Cauchy€™s theorem apply? Show details. 27* + z* + 4 dz, z* + 422 partial fractions. C: |z – 2| = 4 clockwise. Use
Showing the details, developWhat can you say about the coefficients of the Fourier–Legendre series of f(x) if the Maclaurin series of f(x) contains only powers x4m (m = 0, 1, 2, · · ·)?
Show that the integral represents the indicated function. Use (5), (10), or (11); the integral tells you which one, and its value tells you what function to consider. Show your work in detail. w* sin
Find for if if . Try to obtain from it for in Prob. 5 by using (5a).Data from Prob. 5Find f̂c(w) for f(x) = x2 if 0 < x < 1, f(x) = 0 if x > 1.
Are the following functions even or odd or neither even nor odd?Product of an odd times an even function
Show that y" + fy' + (g + λh) y = 0 takes the form (1) if you set p = exp (∫f dx), q = pg, r = hp. Why would you do such a transformation?
Showing the details, develop1, x, x2, x3, x4
Show that the integral represents the indicated function. Use (5), (10), or (11); the integral tells you which one, and its value tells you what function to consider. Show your work in detail. < cos
Find the Fourier transform of (without using Table III in Sec. 11.10). Show details. eka if x 0) if x>0 f(x) f(x) f(w) = F(f) (1 if -b
If f(x) has period p, show that f(αx), α ≠ 0 and f(x/b), b ≠ 0, are periodic functions of x of periods p/α and bp, respectively. Give examples.
Derive formula 3 in Table I of Sec. 11.10 by integration. fe(w) = F.(f) f(x) (1 if 0 0) (See Sec. 6.3.) sin X 11 arctan 12 Jolax) (a> 0) (1 - a(w - a)) (See Secs. 5.5, 6.3.) 2. 3.
Are the following functions even or odd or neither even nor odd?Sums and products of odd functions
Using Prob. 3, derive the orthogonality of 1, cos πx, sin πx, cos 2πx, sin 2πx, · · · on -1 ≤ x ≤ 1 (r(x) = 1) from that of 1, cos x, sin x, cos 2x, sin 2x, · · · on -π ≤ x ≤
ym of (1), (2) means that we multiply ym by a nonzero constant cm such that cmym has norm 1. Show that zm = cym with any c ≠ 0 is an eigenfunction for the eigenvalue corresponding to
Showing the details, develop(x + 1)2
Find the Fourier transform of (without using able III in Sec. 11.10). Show details. if -1
Find the trigonometric polynomial F(x) of the form (2) for which the square error with respect to the given f(x) on the interval -π < x < π is minimum Compute the minimum value for N = 1, 2,
Find f in Prob. 1 from the answer f̂cData from Prob. 1Find the cosine transform f̂c of f(x) = 1 if 0 < x < 1, f (x) = -1 if 1 < x < 2, f (x) = 0 if x > 2.
Are the following functions even or odd or neither even nor odd?sin2 x, sin (x2), ln x, x/(x2 + 1), x cot x
In Example 1, what happens to the amplitudes Cn if we take a stiffer spring, say, of k = 49? If we increase the damping?
Evaluate the integral directly or, if possible, by the divergence theorem. Show details.F = [x, xy, z], S the boundary of x2 + y2 ≤ 1, 0 ≤ z ≤ 5
Evaluate the integral directly or, if possible, by the divergence theorem. Show details.F = [y2, x2,z2], S the portion of the paraboloid z = x2 + y2, z ≤ 9
Evaluate the integral directly or, if possible, by the divergence theorem. Show details.F = [1, 1, 1], S x2 + y2 + 4z2 = 4, z ≥ 0
Evaluate the integral directly or, if possible, by the divergence theorem. Show details.F = [x + y2, y + z2, z + x2], S the ellipsoid with semi-axes of lengths α, b, c
Given a surface S: r (u, v), the differential formwith coefficients (in standard notation, unrelated to F, G elsewhere in this chapter)is called the first fundamental form of S. This form is basic
Why is Ix in Prob. 23 for large h larger than Ix in Prob. 22 (and the same h)? Why is it smaller for h = 1? Give physical reason.Data from Prob. 22Given a mass of density 1 in a region T of
Find the coordinates x̅, y̅ of the center of gravity of a mass of density f(x, y) in the region R. Sketch R, show details.f = 1, R: 0 ≤ y ≤ 1 - x4
If IB is the moment of inertia of a mass distribution of total mass M with respect to a line B through the center of gravity, show that its moment of inertia IΚ with respect to a line K, which is
Given a mass of density 1 in a region T of space, find the moment of intertia about the x-axis Ix = ∫∫T∫ (y2 + z2) dx dy dz.The paraboloid y2 + z2 ≤ x, 0 ≤ x ≤ h
Find the moment of inertia of a lamina S of density 1 about an axis B, where S: x2 + y2 = 1, 0 ≤ z ≤ h, B: the line z = h/2 in the xz-plane
Given a mass of density 1 in a region T of space, find the moment of intertia about the x-axisIx = ∫∫T∫ (y2 + z2) dx dy dz.The ball x2 + y2 + z2 ≤ α2
Find Ix, Iy, I0of a mass of density f(x,y) = 1 in the region R in the figures, which the engineer is likely to need, along with other profiles listed in engineering handbooks. У h- 2. SIN

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