(11 marks] If u(r, y) and e(r, y) are harmonic functions in a domain D, then show that the function dv S(:) - Dy, is analytic in D.

Q: 2. Find the derivative of the following functions: (a). f(r) = r* + 3x 2+ 4e [ %3D (b). f(r) = 2 tan…

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Q: 4. Find the isolates singularities of the following functions, and determine whether they are…

A: "Since you have posted a question with multisubparts, we will solve the first three subparts for…

Q: 5. Find a function u(r, 8) that is harmonic outside the unit disk, equals f(0) = 2 sin(30) + 4 cos…

A: In the given problem we used the fact that f is harmonic on the unit circle, and using the…

Q: One of the following is symmetric about the origin: A) f(x) = x³ + |x]. B) f(x) = x³ cos(x). %3D %3D…

A: If a function is odd then it is symmetric about the origin.

Q: Find the Wronskian of the functions f(t) = 6e sin(t) and g(t) = et cos(t). NOTE: Your answer must be…

A: The given functions are;  f(t)=6etsint and g(t)=etcost.

Q: (sin2z, if 0<zs zz + b,if<zs1" Q4. Let g(x) Derive the values of a and b so that g is continues and…

A: Given, the function g(x)=sin2x,  if  0&lt;x≤π6ax+b,if π6&lt;x≤1. We have to find the value of a and…

Q: 27 Show that ø(x, y)= e*(x cos y –y sin y) is a harmonic function, and find the conjugate harmonic…

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Q: One of the followings is a harmonic function O a. u(x,y)=(x' - y' – 2x+1), O b. u(x,y)=(x-y - 2x+1),…

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Q: Show that u = x³ – 3xy? is a harmonic function and find a corresponding analytic function f(z) = u +…

A: To show that u=x3−3xy2 is a harmonic function and to find a corresponding analytic function fz=u+jv.…

Q: - fyr. For the function f(x, y) = e²´y sin(xy) show that Clairaut's Theorem holds, that is, fæy

A: Clairut's Theorem : fxy = fyx i.e, ∂2f∂x∂y =∂2f∂y∂x.

Q: (a) Sketch the graph of f(t) = sin 2(t- π)U(t - π).

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

Q: Define the hyperbolic functions as follows. e" - e e" + e sinh r .cosh a 2 sinh x and tanh a cosh x…

A: We have to solve both Questions.

Q: Consider the function f on [0,1] defined by f(x) = (t2 – t)dt. Is the function f (x) increasing or…

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Q: (e) intersects the graph z = f(x, y) at P and only at P. Given a function f(x,y) and a point P in…

A: State whether the following statements are true or false. Given a function fx, y at a point P in the…

Q: (2) Find a smooth curve a in GL(2, R) passing through the identity such that a'(0) 1 0 Justify your…

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Q: (a) Use the Product Rule twice to prove that if f , g , and hare differentiable, then (fgh)'   =f'gh…

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Q: #2. Verify Clairaut's Theorem for the given function f (s, t)= In (st)-s* sin (6t) +st.

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Q: Consider the formula f(2) = Log 2+ z (i) For which values of z E C does f define a function of a…

A: Given function is  fz=log2-z2+z             (i) Since logab=loga-logb, (i) can be expressed as…

Q: 2. Find an analytic function f (x+ iy) whose real part is given by the formula u = e* (cos(y) + sin…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Q2: Show that u = ln(x² + y2) is harmonic function and find the conjugate harmonic v = (x. y) and…

A: Given: u(x, y)=lnx2+y2 To show: u(x, y) is harmonic function and find its harmonic conjugate v(x, y)…

Q: (a) Find a harmonic function u on the upper half-plane that has the following boundary values. (1…

A: Detailed explanation mentioned below

Q: 6. Determine whether the following functions are harmonic. (a) u(x, y) = 2e* cos3y and v(x, y) = 2e"…

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Q: Z-T 4. Find all singularities in C for the function f (z) and their residues. sin? z

A: We can solve this question using definition of singularities and residues

Q: 1. Find the derivative of the function at P in the direction of u. Sketch the if possible. c)…

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Q: 4. Show that the following functions are harmonic, and find all harmonic conjugates: a) u = x³ 3xy²…

A: Since you have posted a question with multiple sub-parts, we will solve first three sub parts for…

Q: the function f (2) = z Cos z ̟has a zero of order 1 None

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Q: 2. Verify that the given function find a harmonic conjugate of u. Write an analytic function f (z).…

A: As per our guidelines we are supposed to solve only one question that is the first question. For the…

Q: 3 For which values of a € R the tangent plane to the graph of f(x, y) = 1 + sin(ax + y²) at the…

A: The given graph is                                f(x,y)=1+sin(αx+y2)

Q: Find and simplify the indicated derivatives of the following funetions: (a) f(x, y,z) = (x+1)cos(x²…

A: According to our guidelines, In case of multiple questions we are supposed to answer the first…

Q: of of and da For the function f(t,a) = sin (8rt + 7a), find %3D - at

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Q: Q1] A complex-valued function is given as in the following (z + j)(z + 1) 2. z # 1 f(z) = z2 + 1 %3D…

A: Note:-  1.    j2=-1                2.   z2+1=z+jz-j   Given function is fz= 2z+jz+1z2+1,z≠1  j,z=1…

Q: - [² Suppose F(x)= = sin(t) dt. Use the Fundamental of Theorem of Calculus to calculate - (F(x³)).…

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Q: cos x+sin x-e 2 cos x-2 cosh y If f (z) = u + iv is an analytic function of z and u-v = prove that…

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Q: Let f(x) = 5/2 cos x and let a = 4 Complete parts (a) through (d) below. a. Find the linear…

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Q: 4. Find the isolates singularities of the following functions, and determine whether they are…

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Q: 9–18 Find the directional derivative of f at P in the direction of a. 9. f(x, y) = 4x°y²; P(2, 1); a…

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Q: 1. Find the derivatives of each of the following functions. (a) f(x) = cos x In r O (z) = In(cos a)…

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Q: 2. Consider the function f : R2² → R² defined by f(x, y) = (xe* cos y — ye" sin y, eª y cos y + xe*…

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Q: (26) Let f(x) = x?e=*. For what value(s) of x does f(x) have a horizontal tangent? (a) Find the…

A: ⊙   A curve will have horizontal tangent at a point if its first derivative is zero at that point. ⊙…

Q: Find the points on the interval -n s0s n at which the cardioid r = 1- cos(0) has a vertical or…

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Q: Find the Wronskian of the functions f(t) = 5e* sin(t) and g(t) = et cos(t). NOTE: Your answer must…

A: I have given the answer in the next step. Hope you understand that

Q: One of the following is symmetric about the origin: A) f(x) = x³ + |x). B) f(x) = x cos(x). %3D %3D…

A: Given functions are A) f(x) = x3 + |x| B) f(x) = x3 cos(x) C) f(x) = x5 + 1 D) f(x) = x 3+ x2 E)…

Q: 3 For which values of a E R the tangent plane to the graph of f(x, y) = 1+ sin(ax + y²) at the point…

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Q: Find the nature and location of singularities of the following functions : z - sin z (i) 1 (ii) (z +…

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Q: The conjugate harmonic function v(x, y) of u(x, y) = In (x2 + y2) is *(;) + -1 1. 2tan () + -1 2tan…

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Q: ()- 2y on [–10, 15]. Determine the absolute extrema of f (y) = sin

A: Here we have to find the absolute extrema.

Q: 3 Evaluate F. dr along the curve C if F(x, y) = (x² + y²) ²(xi+yj), and C : r(t) = e' sin ti+ e' cos…

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Q: (11) is harmonic, then find the conjugate harmonic function v(x, y) such that f (z)=u+iv is…

A: The objective is to state whether the function ux,y=lnx2+y2 is harmonic or not. If ux,y=lnx2+y2 is…