Prove that | f(2)dz → ViRes(f, zo) 8 → 0. by letting Is = {z = 20 + de®; 0<0<¥} %3D where f is analytic in a neighbourhood B(zo,r) except for a simple pole at z0.

Q: Let F(x, y, z) = (e" cos y, –e* sin y, 2). Evaluate the integral | F. dr, where C is the curve…

A: Given that, Fx, y, z=excosy, -exsiny, 2 Here, M=excosy, N=-exsiny, P=2. Differentiating M, N and P,…

Q: 2. If f e R[a, b], show using the properties of Riemann Integrability that there exists M > 0 such…

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Q: Use a double integral and Green's Theorem to find $C(x-y)dx+xydy,ke ( - ) dz + ry dy where C C is…

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Q: Use Green's Theorem to evaluate the line integral f(e*c - 2y)dx+(5x+ev* dy, where C is the 15. 2cos:…

A: We need to find integral.

Q: 10. Evaluate the integral cz(z-1) where C is a simple closed curve in the positive direction that…

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Q: Let C, be the curve satisfying |2|=1, C₂ be the curve satisfying |2+1|=12, and C, be the curve…

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Q: Use Green's Theorem to evaluate the line integral: f. In(1 + y) dr – #, dy where C is triangle with…

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Q: Use Stokes's Theorem to evaluate F(x, y, z) = x²i + z²j – xyzk S: z = √√√4x² - y² F. dr. In this…

A: Consider the given function, Fx,y,z=x2i+z2j-xyzk and S:z=4-x2-y2 Let us consider that ϕ:x2+y2+z2=4.…

Q: What is the order of the pole of f(z) = 1 (2 cos z −2+z²)² at z=0? [HINT: Work with 1/f(z).]

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Q: a) Evaluate the line integralZ F dr C in terms of t where F = cosyi+xj+ye²>k and C is the line…

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Q: Evaluate the integral [(e' - y)dx + (sin y +x)dy using Green's theorem where C is 16 bounded by y =…

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Q: Find the local linearization of f(z) =z² at 9. lo(z) = %3D

A: Given : f(x) = x2   x0 =  9

Q: 3. If C is the unit circle in the complex plane C, and f(2)= =², show that f(z) dz = 0 using two…

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Q: a) Evaluate the line integral TM UTM in terms of 7 where F = cos yi+ rj + ye*k and C is the line F…

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Q: Let F = (-z2, 2zx, 4y - x2}, and let C be a simple closed curve in the plane x + y + z = 4 that…

A: Given F={-z2,2zx,4y-x2}. Curve in the plane (s) = x+y+z=4. Area enclosed is 16.

Q: let C denote the polygonal line from (0,0) to (1 + i) evaluate the integral on C f (z) = y - x -…

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Q: Find the sum of the residues of f (z) = sin z at its poles inside the circle |z| = 2. z cos z

A: Residue theorem∮f(z)dz=2πi(R1+R2+.........Rn)             =2πi(sum of the residues at the poles…

Q: (a) Verify that the Fundamental Theorem for Line Integrals can be used to evaluate the integral Sc…

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Q: A- Find all values of z such that sinh(z) = B- Are |z, Img(z) analytic? Give reason. C- Choose the…

A: We have to solve the given questions

Q: Let C denote the line segment from the point -2+i to the point 1+7i and let f(2) where z = x + iy.…

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Q: Verify the Green's theorem for integral x²y dx + xy dy, where C is the boundary described…

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Q: Suppose that (z) is entire and that the harmonic function u(x, y) = Re[fz)] has an upper bound uo;…

A: As requested following is the answer for question 12.

Q: 2. Evaluate fy³ dx-r³ dy, where C is the circle of radius 2 centered at the origin, in the following…

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Q: (a) Find the line integral of f (x, y, z) = -x + y + z over the straight-line %3D segment from (1,…

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Q: Evaluate the line integral F. dr, vhere F(z, y) = -yi-rj and C is the curve y = v4-a from (2,0) to…

A: We have to find the line integral

Q: Use Green's Theorem to evaluate the line integral F.dr, where F(x, y) = (x√√x+y₁x² + √√) and C is…

A: Here we have to apply greens theorem.

Q: Use the Green's Theorem to evaluate the line integral 2xydx + x² – y² dy where C is the curve…

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Q: Q7. Use Green's theorem to evaluate fy² dx + 3xy dy, where c is the boundary of the semi annular…

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Q: find the holomorphic function f(z) = u(x,y) + j*v(x,y)  if  v(x,y) = -ey * sin(x-y), f(0) = 1

A: The general form of a complex number is given by z=a+jb, where a,b are two real numbers and j=-1.…

Q: Evaluate both sides of divergence theorem for A= pa,+ pcos()a+ za, through the closed surface…

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Q: Use Green's Theorem to evaluate / F · dĩ, where F = (VE + 3y, 2x + vũ) C consists of the arc of the…

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Q: (b) Let F(x,y) = (e2*y+cos(7x²), xºy^+e>Y³} Use Green's Theorem to set up the line integral F. dr as…

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Q: Evaluate the line integral | F•ds where F = 2xyi +(x – y²)j and C is (a) y=x from (0,0) to (1,1) (b)…

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Q: d. Dse the bosic properhes of integrC followng theorem. THEOREH:If (Cx. Y) = FG)G(), wbe La,bland G…

A: We will use dominated convergence theorem to prove the given theorem.

Q: 2. Compute the scalar line integral xy ds where C is the part of the circle of radius 2, centered at…

A: Put x=2cost , y=2sint

Q: Let R:x²<ys 3 be the closed region having boundary C with counter-clockwise orientation. F=[X…

A: We are given that the closed region R: x2 ≤ y ≤ x/3, is having boundary C with counter-clockwise…

Q: Use the Green's Theorem to evaluate the line integral f, FdT where F = (sin(x²)-2x²y) i+ (cos(y?) +…

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Q: Suppose C is a positively oriented, piecewise-smooth, simple closed curve. Verify that the result of…

A: Given    ∫Cxy2dx-x2ydy Where C is arc of parabola y=x2 from -1,1 to 1,1 and the line segment 1,1 to…

Q: Use Green's theorem to evaluate the closed line integral: . x²y dx +x'y dy along C between -1 < x< 2…

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Q: Evaluate the line integral: f y² dx + x² dy by parameterizing C where the curve C is a square R with…

A: According to the given information, it is required to evaluate the line integral.

Q: -. Let F = (1 + xe-Y)i -e-j. Show that F is conservative and, hence, use the Fundamental Theorem for…

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Q: Use Stokes's Theorem to evaluate F dr, where F(x, y, z) = x²i+z²j-xyzk and S is z = /169-x² -y² .…

A: As per our guideline we can't use any computer algebra. I calculate both line integral and surface…

Q: b. What is fo, f (z) dz?

A: f(z) is holomorphic except at z0   where f(z) has pole at z0 we can write  f(z)=g(z)z-z0  where g(z)…

Q: Verify Green's theorem in the plane for (3x² -8y²)dx+(4y-6xy)dy, where C is the closed curve of the…

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Q: Evaluate the line integral [F-dš where F = 2xyi + (x – y')j and C is (a) y =x from (0,0) to %3D…

A: For the solution follow the next steps.

Q: 2. Find the parametrization of the tangent line to the space curve r(t) = (-t3, In(t), e² ) at t =…

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