Question

Find the value of A, B, C, D, and E

Let f (z)be an analytic function, which is represented by f (x) = u(x, y) + iv (x, y) where z = x+iy If u(x, y) = 38x (49 – y) - 42 and v(x,y) = Ax? +By? + Cx + Dy+ E and v(0,0) = 45, then

Answer 1

To find the values of A, B, C, D, and E, we need to use the Cauchy-Riemann equations, which state that if f(z) is analytic, then:
u_x = v_y u_y = -v_x
where u_x and u_y are the partial derivatives of u with respect to x and y, and v_x and v_y are the partial derivatives of v with respect to x and y.
Using these equations, we get:
u_x = 38(49 - y) = v_y u_y = -38x = -v_x
Therefore,
v_y = 38(49 - y) v_x = 38x
Integrating both sides of the first equation with respect to y, we get:
v(x,y) = 38(49y - y^2/2) + g(x)
where g(x) is an arbitrary function of x.
Integrating both sides of the second equation with respect to x, we get:
v(x,y) = 38x^2/2 + h(y)
where h(y) is an arbitrary function of y.
Equating these two expressions for v(x,y), we get:
38(49y - y^2/2) + g(x) = 38x^2/2 + h(y)
Comparing the coefficients of x^2, y^2, xy, x and y, we get:
g’(x) = 0 h’(y) = 0 0 = 0 g(x) = 19x^2 + C h(y) = -19y^2 + D
where C and D are constants.
Using the given condition that v(0,0) = 45, we get:
v(0,0) = g(0) + h(0) 45 = C + D
Therefore,
C + D = 45
Hence, we have found that:
v(x,y) = 19x^2 - 19y^2 + Cx + Dy + E
where A = 19, B = -19, C and D are constants that satisfy C + D = 45, and E is another constant.