1. Verify whether the function f(z) = e-Y sin x – ie- cos x is entire or not. 2. Show that the function f (2) = xy + iy is nowhere analytic. %3D 3. Show that u(x, y) = 2x – x³ + 3xy? is harmonic. 4. Find the harmonic conjugate of u where u(x, y) = x2 – y² – 2xy – 2x + 3y. sinh 4x cos 4y + i cosh 4.x sin 4y satisfies 5. Verify whether the function f(2) Cauchy-Riemann equations or not.

Q: Find an analytic function whose real part is u(x, y) = x³y - y³x 7²+1

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

Q: a) Find the points at which the function f: C → C defined as: f(x + yi) = x³ − y² + (x − y)i - Have…

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Q: Q1) Verify that the following functions u is harmonic, and find a conjugate harmonic of function…

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Q: - Let f(x, y) =sin(x y − x − 3y+3). (vi) Which of the surfaces below is a graph of this function?

A: Given .This can be rewritten asOn the xy−plane, , if  gives us hyperbolas on the xy−plane.

Q: Consider the function z= = f(x, y) = sin(x - y). (a) Plot the cross-section where the graph of z =…

A: Since you have posted multiple questions with multiple sub-parts, we will provide the solution only…

Q: The conjugate harmonic function v(x, y) of u = In (x² + y²) is

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Q: 4 - Find the direction A in which the function f(x, y) = x cos y + y sin x decreases most rapidly at…

A: We have to find the direction A in which the given function decreases most rapidly.

Q: Determine the order, degree, unknown function, and the independent variable of the DE 2x + 3x – 5x =…

A: in the first problem  order = 3 degree =1 unknown function =x(t) unknown variable = x independent…

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Q: Differential Calculus:

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Q: 2.(2.3) Let f(x, y) = xy(e² + e³) for (x, y) = R². (1) Find fx and fy. (2) Find an equation of the…

A: (1) fx​=yex+yey+xyexfy​=xex+xey+xyey (2)(3e)x+(3e)y−z=4e  Explanation:

Q: Consider the function z = f(x,y) = sin(x – y). (a) Plot the cross-section where the graph of z =…

A: Given z=sinx-y. (a) The xz plane is given by y=0. We get, z=sinx. This is just the sine curve in the…

Q: Show that u(x, y) = 2x(1 - y) is harmonic and find a harmonic conjugate v(x, y). Show your solution.

A: ux , y= 2x1-y To Find: Show that ux , y= 2x1-y is harmonic and also finds a harmonic conjugate.

Q: Given the complex function f(z) ee-iy. Where are ƒ(z) differentiable? Where is it holomorphic?…

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Q: Verify that u(x, y) = x² + 4x – y² + 2y is harmonic and find %3D - a) the harmonic conjugate…

A: The solution is given as

Q: Find The artlal Auatives (first and Seond

A: Since the student has posted multiple questions and does not mention any specific question, so we…

Q: 9. Let f(x) = x sin²x, g(x) = x cos x, and h(x) = x. Then the functions f, g, and h are linearly…

A: The given functions  and .The aim is to check whether the functions are linearly independent or not.

Q: on f(z) = e cos x+ ie-Y sin x is entire.

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Q: Find a so the function u(x, y) = e2" cos 2ay + a²x – y + 15 is harmonic. Then, find the complete…

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Q: Find a harmonic conjugate v(x, y) of u(x, y) = e e sonding analytic function ƒ(z) = u(x, y)|+ iv(x,…

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Q: 1. Let f: R²R where f(x, y) = x² + y² and c = (2, 2). (a) Find and simplify Df(c). (b) Find and…

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