5. Let C be the unit square with diagonal corners at –1- i and 1+i. Evaluate O f(z) dz, where f(z) is given by the function sin z.

Q: Suppose z = x + 2i and u= 3+ yi where x, y E R. Find the smallest positive value of x z + u for…

A: Substituting z=x+2i and u=3+yi in z+uz-u:  z+uz-u=x+2i+3+yix+2i-3+yi           =x+3+2+yix-3+2-yi

Q: 2. Consider the PDE u; - t ux+tu=0 for - o <x< ∞ and 0 <t< ∞ with u (0, x) = sine x. Determine u (t,…

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Q: Determine fry when f (x, y) = x sin¬' () -1 1. fry = (x² – y?)3/2 y? x² – y? 2. fry - xy 3. fry (x²…

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Q: Consider f(x,y) = cos(x² e³). Then which one of the following is correct O!. fx(x, y) = – 3 y² sin(3…

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Q: The function yow) satisfies the differantial equation x dy + ay Cos(xx) ak y(1) = 1 , the valu of…

A: Given differential equation is  xdydx+2y=cos(πx)x,y(1)=1

Q: If f(x,y,z)=sin(xy)+x2lny, then fxy at (0,pi/2) is equal to 1 FO

A:

Q: If y is implicitly defined as a function of e by –2 + + cos(y) = 3, determine de ysín(oy) + 2 -1…

A: We have to find dy/dx.

Q: If y = cos ¹ prove Zyy + z = 0. )Z = (1 + i)(2-1)-¹, find |Z|.

A: Here given y=cos-1(z/x). We prove that zyy+z=0.

Q: By the odd extension method applied to the function f(x)=e-2=, where r > 0, the Fourier integral of…

A: Given function fx=e−2x where, x&gt;0 We have to find the Fourier integral by the odd extension…

Q: in term of t, if w -+, x= cos t,y sint and z in term of t. if w = xy, x = cost and y = sint, whent=.…

A: 4. Given data is : w=xz+yz , x=cos2t , y=sin2t  and z=1t To express dwdt in terms of t

Q: 2. (a) Let f(x, y) = x² sin(x + y), find 잃 and 잃.

A:

Q: 1. If y= tan x, prove that y"= 2(y² +1)(3y² +1).

A:

Q: az Suppose z is a function of r and y, and csc Vr² + yz = 2"e»in z_ Solve for dz and dy

A: Given: csc x2+yz=zyesin x To find ∂z∂x and ∂z∂y.

Q: Solve the D. E. to the I. V. P. sinx(e + 1)x'- cosx 1, y() = 0 O (1 + e') (1 + cos x) = 2 O (1 + e)…

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Q: 1. If y= tan x, prove that y``= 2(y² +1)(3y² +1).

A:

Q: 3. Let G = {z = x+iy : x < 0 or y # e" sin x}. Show that G is simply connected.

A: Given: G=z=x+iy; x&lt;0 or y is not equal to ex sin x

Q: 3) Calculate Wronskian and evaluate for which values a, b ER ne following sets of functions are…

A: Given functions are: 1) sinax,cosbx 2) sinhax,coshbx Calculate the Wronskian and evaluate for which…

Q: - If x=sect and y = tan t, evaluate at t of the following: 4. dy (a) dx d y (b) dx?

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Q: For f (x) = sin x + 2, there exists a 'c' in [0, 27], Which satisfies the conditions of Rolle's…

A: From the given problem:   fx=sinx+2 ;0,2π   For verify the Rolle's theorem:   1) f(x) is continuous…

Q: Consider the function. X 10/17 - 12/24 5 Find Duf(7, 1), where u = cos(0)i + sin(0)j. f(x, y) = 8

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Q: 8. Verify that the function u(x, y) = eª sin(y) is harmonic in the region where it is defined and…

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Q: əz Əz Əx Əy Q5-A) If z =(x +by)² +e+by+sin( ax +by) show that b =a

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Q: 4. Let z = x2 + y³, x = 2st, y = s - t2, s = u", and t = u sin w. Compute for Oz ду and дw ди

A: Let z=x2+y3, x=2st, y=s-t2, s=uw, and t=u sin w.To compute for ∂z∂w and ∂y∂u.

Q: . Solve the following problems: (a) Utt = 0, 0 0, u (x, 0) = x + sin x, Ut (x, 0) = 0, 0 0.

A: Solution

Q: 2. (a) Determine c, and cz that y = c¿e³x + c2e¯2x + 2 sin x will satisfy the conditions y(0) = 0…

A: We have to determined the value of C1 and C2that y=C1e3x+C2e-2x+2sinx will satisfy the condition…

Q: Suppose that the joint p.m.f. of Y and Y2 is defined by: 2x + y for x = 1, 2,3; y = 1,2,3 54 f(x, y)…

A: The probability mass function of Y1 and Y2 is,

Q: 1. (§5.3, Q6) Find the general antiderivative of f and check your answer f (x) = 4x' – 3 cos x. by…

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Q: Given f(x,y) = eX cos y, Evaluate fy(2,0) O cos 2 O sin 2

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Q: 1)the f a) 3 cosh z b) 0 c) in d) n -dz by using the residues is equal to

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Q: Q:Ifx= sin 6, y = 0,y = cos pe, prove that (1-x )y,-xy,+p°y = 0, %3D dy where y, dy and y1 d

A: Here given that x=sinθ, y=cospθ Taking derivative of both x &amp; y with respect to θ, we get…

Q: Q9. If W = f(u)=u³ +sin ¹u+lnu, and u= ax +by, where a, and b are constants, show that aw Əw b oy Əx

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Q: 6. Suppose f(2) is analytic in |2| 1, and f(0) = 1, f(0) = 2, compute: %3D (1) h = fe(s+ 1)2d: (2)…

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Q: Given that (1 COS 4.r) x2 a, if x = 0 if x 0. J 16+ Vx -4 If f(x) is continuous at x = 0, find the…

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Q: - Show that u(x,t) e-a't[A sin(2x) + B cos(Ax)] satisfies the PDE uz = a?uxx for arbitrary A, B and…

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Q: 1. For the function (x, y) = x²y + 2xy² + sin(xy), compute fx , fy , fxx fyy , and fry

A:

Q: 15. Determine if the given functions are linearly dependent or independent. (Prove your answers) {e…

A: 15.  a) Given, e2x, 3x Now the Wronskian is, Wf, g=fxgxf'xg'x=e2x3x2e2x3=3e2x-6xe2x≠0 Since the…

Q: If f(x) = (e"1+z)' 15 sin(z) +8 cos(z) f'(0) then f(0)

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Q: Analyze the approximate value and the exact value for (7.2, 4.06) for the function z = x3y2 sin(xy)…

A: Given below clear explanation

Q: Find the and of the following function; дх ду (a) z = xln(xy) + y³, y= cos(x² + 1) (b)z = V x² +…

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Q: Suppose r E C[0, 1]. Show that there exists unique y E C[0, 1] such that x(t) = y(t) + 2 5 sin(s +…

A: Consider the provided question,

Q: If f(x,y) = sin(xy) +x² In(y) .Find fyx at (0,1/2) %3D а. 1 b. -1 с. О d. none of the above

A:

Q: So Evaluate F. dr where C F = (– 2z, – y, – 4x), and C is given by 7(t) = (sin(t), cos(t), t), 0 < t…

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Q: dz is given by ди Let z= sinh -(y - 2x), where x = we e-3u and y= In then - OA. 6we -3u Vi- (y- 2x)?…

A: Here we will use chain rule of differentiation. and d/dx(sinh^(-1)(x)) =1/√(1+x^2)

Q: Suppose z = x + 2i and u = 3+ yi where x, y E R. Find the smallest positive value of x z + u for…

A: Given that z=x+2i and u=3+yi where x,y∈R. Now our aim is to find the smallest value for x for which…

Q: Analyze the approximate value and the exact value for (7.2, 4.06) for the function z = x³y? sin(xy)…

A: Given below clear explanation

Q: [6 Let f(x) = |r| be approximated by sN(x) = }a0+E (an COS (") + b, sin (7). %3D n=1 Determine the…

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Q: 2,5 Given f(z, y) sin(x + y) where 2s 5t. Find f.(ar(s, t), y(s, t)) f(x(s, t), y(s, t)) =| Note:…

A: It is given that f (x, y) = sin (x + y) where x = s2t5 and y = 2s–5t.

Q: 1 Use the substitutions given to find fI+x dx; u = 1 + x + sin x dx; u = sin x cos x a

A:

Q: Let f(z) = sin(3z) and C: |z| = 1. Then ſ. f(z)dz = z2(z-n) а. 0 b. Зпі с. —6пі d. None of the above

A: Solution: Given fz=sin3zz2z-π and C : z=1 We have to calculate ∫Cfzdz