3. Let u(x, y) be a nonconstant harmonic function in the disk D= for each 0< r < R {a?+y? < R*}. Define M(r) = max u(x, y). x²+y²=r2 Prove that M(r) is a monotone increasing function in the interval (0, R).

Q: What is the domain of the following function? f(z,9) 4-2 - y - In(y - z) + sin(z) +y 24 and y> =)…

A: Domain.

Q: 3. Find the fixed point(s) of f : [1, 0) → [2, 0) defined by f(x) = vx² + x.

A:

Q: Let f be a function from N to (0,1). Prove -h for any nEN, y = f(n). Using part (a), prove that…

A:

Q: Let f(z) be an analytic function, which is represented by f(z)=u(x,y)+iv(x,y) where z=x+iy . If…

A:

Q: ind positive constants b and n, such that 3n + 2 n.

A: Finding the values of b and no

Q: 8. Let g(r, 9) = (a) Show that g is harmonic in D- {(7,9) : + (y +3)2 < 4}. (b) How can the funetion…

A: Here given as gx, y=yx2+y2 ......(1)

Q: Determine the largest subset S of C where the function f(z) = 3x2 + 2x - 3y2 - 1 + i(6xy + 2y) is…

A:

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

A: See the attachment

Q: Let r(t) = (sin(t), cos(t), 6 sin(t) + 9 cos(2t)). Find the projection of r(t) onto the xz-plane…

A: Given vector…

Q: Prove that B(m, n) = B(m + 1, n) + B(m,n + 1)

A: given,

Q: Prdlem 6.36 Pert B)

A: It is double Integral

Q: Define a harmonic function. (b)Prove whether u(x, y) = x² - 2xy + y² is a harmonic function on a set…

A:

Q: The graph of the function below intersects at t axis on the interval [0, 4rt] at points f(t) = cos t…

A: We know that f(t) = cos(t) It cuts the t axis at the points π2,0 , 3π2 , 0 , 5π2 , 0 , ...

Q: 5. Let f(z) = 3z* + iz3 + z + 7. Prove that all roots of f (z) are in the annulus 1 < |z| < 2.

A:

Q: anction of z = x + iy

A:

Q: af Ꭷf f(x, y) — f(y, x) = (x − y) — (x, y) + (y − x) — (x², y") ду or some point (x*, y*) € R² b)…

A: aSince given partial differential equation is               fx=fy⟹      fx-fy=0Auxillary equation…

Q: The function f(x) = |ax - b| + c|x| : x in (-∞, ol where Find the

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Q: 2) Find the complex Founer senes of a 2c - periodic function f, which is defined as f(x)= (ix-1)² in…

A:

Q: 4) Graph y = arcsin x and z = arcsin (x-2) and w= arcsin (x/2) on the same axes Remember that these…

A:

Q: Calculate the divergence of cach of the following functions (a) r²î + y²§ + 2²k. (b) yzî + xzj+ xyk.…

A: As per the instruction, we will answer only the first 3 subparts. For (a), For a given vector fx, y,…

Q: 3. Consider the function f (x) = 2 x + 1 on the interval [2, 6]. (a) Compute the left-endpoint…

A: Solve the following

Q: 5. Determine whether the closed interval [a,b]is homeomorphism to the closed unit…

A: This is a problem from topology.

Q: y), the harmonic conjugate of u. Form the

A: A function u (x, y) is said to be a harmonic function if u _x x + u_ y y = 0.

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

A:

Q: -y If u=x² - y² and v= x* +y then A) u is a harmonic function B) v is a harmonic function C) f(z) =…

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Q: Let f(z) : D → C be a complex function. Is it possible for both f(z) and f(z) to be analytic? • If…

A:

Q: Consider the relation x2z2 = arctan (xyz) + 4, which defines z as an implicit function of x and y,…

A:

Q: y3 be a harm C conjugate. D[ is equal to

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Q: Let f and g be complex valued function defined on [a,b] and let f and g be differentiable at a point…

A:

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.