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advanced mathematical methods for scientists and engineers solutions manual

Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 1 pdf

Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 1 pdf

Kĩ thuật Viễn thông

... 156632.7.3 Cosine and Sine Transform in Terms of the Fourier Transform . . . . . . . . . . . . . . . . . . 156832.8 Solving Differential Equations with the Fourier Cosine and Sine Transforms . . . ... = p(x)/q(x) where p(x) and q(x) are rational quadraticpolynomials. Give possible formulas for p(x) and q(x).Hint, Solution12 33 The Gamma Function 160533.1 Euler’s Formul a . . . . . . . ... also contains the word Scientists or Engineers ” the advanced book may be quite suitable for actually learning the material.xxvii 43.3 The Method of Characteristics and the Wave Equation ....
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Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 2 ppt

Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 2 ppt

Kĩ thuật Viễn thông

... is the angle from a to b and n is a unit vector that is orthogonal to a and b and in the direction s uch thatthe ordered triple of vectors a, b and n form a right-handed system.29 abbθbFigure ... xzyjikzkjiyxFigure 2.7: Right and left handed coordinate systems.You can visualize the direction of a ì b by applying the right hand rule. Curl the fingers of your right hand in thedirection from ... arbitrary vectors a and b. We can write b = b⊥+ bwhere b⊥is orthogonal to a and bisparallel to a. Show thata ì b = a ì b.Finally prove the distributive law for arbitrary b and c.Hint...
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Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 3 pptx

Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 3 pptx

Kĩ thuật Viễn thông

... δ −√x is a decreasing function of x and an increasing function of δ for positive x and δ. Bound this function for fixed δ.Consider any positive δ and . For what values of x is1x−1x + δ> ... Consider y = x3 and the point x = 0. The function is differentiable. The derivative, y= 3x2ispositive for x < 0 and positive for 0 < x. Since yis not identically zero and the sign ... an> 0 for all n > 200, and limn→∞an= L, then L > 0.4. If f : R → R is continuous and limx→∞f(x) = L, then for n ∈ Z, limn→∞f(n) = L.5. If f : R → R is continuous and limn→∞f(n)...
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Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 4 pptx

Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 4 pptx

Kĩ thuật Viễn thông

... of x and an increasingfunction of δ for positive x and δ. Thus for any fixed δ, the maximum value of√x + δ −√x is bounded by√δ.Therefore on the interval (0, 1), a sufficient condition for ... −2)−1/3The first derivative exists and is nonzero for x = 2. At x = 2, the derivative does not exist and thus x = 2 is acritical point. For x < 2, f(x) < 0 and for x > 2, f(x) > 0. ... satisfying,e(x, δ) ≤ (δ), for all x in the closed interval. Since (δ) is continuous and increasing, it has an inverse δ(). Now note that|f(x) − f(ξ)| <  for all x and ξ in the closed interval...
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Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 5 pdf

Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 5 pdf

Kĩ thuật Viễn thông

... |r(t)|.Differentiation Formulas. Let f(t) and g(t) be vector functions and a(t) be a scalar function. By writing outcomponents you can verify the differentiation formulas:ddt(f · g) = f· ... Figure 5.12.) We find the volume obtained by rotating the172 Set x = 2 and x = −2 to solve for a and b.Hint 4.16Expanding the integral in partial fractions,x + 1x3+ x2− 6x=x + 1x(x ... −14sint2j.See Figure 5.8 for plots of position, ve locity and acceleration.Figure 5.8: A Graph of Position and Velocity and of Position and AccelerationSolution 5.2If r(t) has constant speed,...
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Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 6 pps

Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 6 pps

Kĩ thuật Viễn thông

... two, one or no solutions. For example:ã x2 3x + 2 = 0 has the two solutions x = 1 and x = 2.ã For x2− 2x + 1 = 0, x = 1 is a solution of multiplicity two.ã x2+ 1 = 0 has no solutions. 180 ... y2eı arctan(x,y).Cartesian form is convenient for addition. Polar form is convenient for multiplication and division.Example 6.3.1 We write 5 + ı7 in polar form.5 + ı7 =√74eı arctan(5,7)We ... u0, u1, u2 and u3are real numbers and ı,  and k are objects which satisfyı2= 2= k2= −1, ı = k, ı = −k and the usual associative and distributive laws. Show that for any quaternions...
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Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 7 ppt

Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 7 ppt

Kĩ thuật Viễn thông

... real-variablecounterparts.7.1 Curves and RegionsIn this section we introduce curves and regions in the complex plane. This material is necessary for the study ofbranch points in this chapter and later for contour integration.Curves. ... function, f(z) = z. In Cartesian coordinates and Cartesian form, the functionis f(z) = x + ıy. The real and imaginary components are u(x, y) = x and v(x, y) = y. (See Figure 7.9.) In modulus-2-1012x-2-1012y-2-1012-2-1012x-2-1012x-2-1012y-2-1012-2-1012xFigure ... arctangent that is between0 and π. The domain and a plot of the selected values of the arctangent are shown in Figure 7.8.CONTINUE.7.4 Cartesian and Modulus-Argument FormWe can write a function...
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Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 8 ppt

Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 8 ppt

Kĩ thuật Viễn thông

... inCartesian form and z = reıθin polar form.eu+ıv= reıθWe equate the modulus and argument of this expression.eu= r v = θ + 2πnu = ln r v = θ + 2πnWith log z = u + ıv, we have a formula for ... SeeFigure 7.18 and Figure 7.19 for plots of the real and imaginary parts of the cosine and sine, respectively. Figure 7.20shows the modulus of the cosine and the sine.The hyperbolic sine and cosine. ... 11/2.Construct cuts and define a branch so that z = 0 and z = −1 do not lie on a cut, and such that f(0) = −ı. What isf(−1) for this branch?Hint, Solution290 The only solutions that satisfy...
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Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 9 ppt

Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 9 ppt

Kĩ thuật Viễn thông

... HintsCartesian and Modulus-Argument FormHint 7.1Hint 7.2Trigonometric FunctionsHint 7.3Recall that sin(z) =1ı2(eız−e−ız). Use Result 6.3.1 to convert between Cartesian and modulus-argument form.Hint ... on which f(0) = ı√6. Write out an explicit formula for thevalue of the function on this branch.Figure 7.33: Four candidate sets of branch cuts for ((z − 1)(z − 2)(z − 3))1/2.Hint, Solution294 ... 4.Figure 7.48 first shows the branch cuts and their s tereographic projections and then shows the stereographic projectionsalone.Solution 7.211. For each value of z, f(z) = z1/3has three...
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Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 10 doc

Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 10 doc

Kĩ thuật Viễn thông

... condition for the analyticity of f(z).Let φ(x, y) = u(x, y) + ıv(x, y) where u and v are real-valued functions. We equate the real and imaginary partsof Equation 8.1 to obtain another form for the ... ı)−1/3=3√reıθ/313√se−ıφ/313√te−ıψ/3=3rsteı(θ−φ−ψ)/3we have an explicit formula for computing the value of the function for this branch. Now we compute f (1) to see if wechose the correct ranges for the angles. (If not, we’ll just ... . . 2π), (2π . . . 4π), . . .}.Now we choose ranges for θ and φ and see if we get the desired branch. If not, we choose a different range for one ofthe angles. First we choose the rangesθ ∈...
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Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 1 pps

Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 1 pps

Kĩ thuật Viễn thông

... equations for à and are satised if and only if the Cauchy-Riemannequations for u and v are satisfied. The continuity of the first partial derivatives of u and v implies the same ofà and . Thus ... =x3(1+ı)−y3(1−ı)x2+y2 for z = 0,0 for z = 0.Show that the partial derivatives of u and v with respect to x and y exist at z = 0 and that ux= vy and uy= −vxthere: the Cauchy-Riemann ... function.Solution 8.11We write the real and imaginary parts of f(z) = u + ıv.u =x4/3y5/3x2+y2 for z = 0,0 for z = 0., v =x5/3y4/3x2+y2 for z = 0,0 for z = 0.The Cauchy-Riemann...
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Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 2 pptx

Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 2 pptx

Kĩ thuật Viễn thông

... Consider analytic functions f1(z) and f2(z) defined on the domains D1 and D2, respectively. Suppose that D1∩ D2is a region or an arc and that f1(z) = f2(z) for allz ∈ D1∩ D2. (See ... converges uniformly for D1= |z| ≤ r < 1. Since the derivative also converges in this domain, the function isanalytic there.440 Figure 8.7: The velocity potential φ and stream function ψ for Φ(z) ... Substitute this expression for v into the equation for ∂v/∂x.−ye−xsin y − xe−xcos y +e−xcos y + F(x) = −ye−xsin y − xe−xcos y +e−xcos yThus F(x) = 0 and F (x) = c.v =e−x(y...
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Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 3 ppt

Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 3 ppt

Kĩ thuật Viễn thông

... contour and do the integration.z − z0=eıθ, θ ∈ [0 . . . 2π)C(z − z0)ndz =2π0eınθıeıθdθ=eı(n+1)θn+12π0 for n = −1[ıθ]2π0 for n = −1=0 for n = −1ı2π for ... uy) dx dy + ıD(ux− vy) dx dy= 0Since the two integrands are continuous and vanish for all C in Ω, we conclude that the integrands are identically zero.This implies that the Cauchy-Riemann ... ıφ dy)=D(ıφx− φy) dx dy= 0Since the integrand, ıφx− φyis continuous and vanishes for all C in Ω, we conclude that the integrand is identicallyzero. This implies that the Cauchy-Riemann...
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Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 4 ppsx

Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 4 ppsx

Kĩ thuật Viễn thông

... converges if and only i f for any  > 0 there exists anN such that |an− am| <  for all n, m > N. The Cauchy convergence criterion is equivalent to the definition we hadbefore. For some ... integrals along C1 and C2. (We could alsosee this by deforming C onto C1 and C2.)C=C1+C2We use the Cauchy Integral Formula to evaluate the integrals along C1 and C2.C(z3+ ... integralCeztz2(z + 1)dz.There are singularities at z = 0 and z = −1.Let C1 and C2be contours around z = 0 and z = −1. See Figure 11.6. We deform C onto C1 and C2.C=C1+C2520 11.4 ExercisesExercise...
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Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 5 pps

Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 5 pps

Kĩ thuật Viễn thông

... ,converges for α > 1 and diverges for α ≤ 1.Hint, Solution564 Example 12.3.2 Convergence and Uniform Convergence. Consider the serieslog(1 − z) = −∞n=1znn.This series converges for |z| ... necessarycondition for uniform convergence. The Weierstrass M-test can succeed only if the series is uniformly and absolutelyconvergent.Example 12.2.1 The seriesf(x) =∞n=1sin xn(n + 1)is uniformly and ... Thus this series is not uniformly convergent in the domain|z| ≤ 1, z = 1. The series is uniformly convergent for |z| ≤ r < 1.545 12.2.2 Uniform Convergence and Continuous Functions.Consider...
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