(iii) af A function is analytic if = 0. Prove that f(x, y) = az x-1-ty (x-1)²+y² is analytic.

Q: 2. Use the First Fundamental Theorem of Calculus to differentiate the function f(r) = VI+ tª dt.

A: We will use the first fundamental theorem of calculus to evaluate the derivative.   The detailed…

Q: Is this function analytic fizz=. (1 +x)? +y2 2

A:

Q: „Show that the complex variable function below is differentiable analytic only at origin point? f(z)…

A: We will show the required result.

Q: Q3-A) Is the function F(Z) continuous or not at z-U? 1+|21,

A: Let z = x+iy then Re(z) = x and |z| = √(x^2 + y^2) If z = 0 => x = 0 and y = 0

Q: Find the domain of f(z) the analytic function, graph on the z-plane.find the Re(f'(1)). f(z) = (z² +…

A: Given is the function f(z)=z2+1i. We need to find the domain of the function.

Q: Find the solution when r = 4 according to the theorem below

A: We have to find the required solution for the given values when r = 4.So for this we put r=4 in…

Q: Examine whether the following function is Analytic or not ¹() (z) = ln(x² + y²) + i tan-¹

A:

Q: c) Consider the third-order truncation error (0 (Ax )³) with the step size Ax and also (-Ax ) then,…

A: Both approximation is second order O((∆x)^2)

Q: Q2/Show that the functions is harmonic u = 2 x (1 - y)

A:

Q: Prove that the function (z*)2 1리 > 0 0, z = 0 f (z) = is not analytic at z = 0 by calculating all…

A: Calculate the derivative of fz at z=0.  f'0=limZ→0fz-f0z-0       =limZ→0z*2/z-0z-0       =limZ→0z*z2

Q: We can not find an analytic function f=u+vi such that 3u-v is not harmonic. * True False

A:

Q: Discuss the analyticity of  f(z) = Re(z 3)   Find whether the given function is analytic or not.…

A: A complex valued complex variable function is analytics at z0 if it is differentiable every point in…

Q: Use Green's Theorem to evaluate F = (xy + y²)i+ (x – y)j | (1, 1) x = y² y = x² (0, 0)

A: we have to evaluate the following by using greens theorem

Q: Show that f(x) = 1/(1+x) is analytic on (-1, ∞).

A: To show that f(x) = 1/(1+x) is analytic on (-1, ∞), we need to show that f(x) has derivatives of all…

Q: Find the analystic function f(z)u+iv if u=a(1+cos0)

A:

Q: (6) If f(z) is analytic function show that: [v] + [0] =

A:

Q: Find all points at which f(z) = (x)/(x2+y2) - i(y)/(x2+y2) is differentiable. At what points is f…

A:

Q: Prove that the function n = x(x^2 - 3y ^2)+ (x^2 - y^2) + 2xy is harmonic. Also find the conjugate…

A: Then,     and Thus,   and , it follows  that .Hence, the  given function is harmonic.(Proved)…

Q: Let f : D → E be a function and let A C D, BC E. Prove the following: (a) f(f-1(B)) C B. (b) If f is…

A: The function is f:D→Eand A⊆D and B⊆E

Q: Construct an analytic function of the form f (z) = u + iv, where v is tan- (y/x), x # 0, y # 0

A: f(z)=u+iv where v=tan-1(yx)  We need to find the analytic function f(z)=u+iv  we apply Milne Thomson…