III SEMESTER B.E, EC, EE, BME, ICEII-TUTORIAL

Part – A

1. Show that f(z) = zn is analytic for positive integral values of n and hence find f (z).

2. Are the following functions analytic? (i) f(z) = Re z or Im z

(ii) f(z) =

(iii) f(z) = z +

3. If f(z) is analytic in a domain D and |f(z)| = const in D, show that f(z) = const.

4. If u (x,y) and v(x,y) are harmonic functions in a domain D, then prove that

is analytic in D.5. If f(z) = u+iv is an analytic function of z =r ei , prove that both u and v

satisfy

6. Prove that u = e – x (x cosy + y siny) is a harmonic function. Hence, find

such that = u +iv is analytic sucht aht = 1 at the orgin.

7. If f(z) = u + iv is analytic function of z , show that

(i)

(ii)

8. If f(z) = u +iv is an analytic function of z, find f(z) if

u – v = (x – y) (x2 +4xy+y2).

9. Show that the following functions are harmonic and find the corresponding analytic function f(z) = u + iv and its conjugate harmonic function

10. Find the analytic function f(z) = u + iv for which

(i) (ii) u + v =

Part – B

1. Prove that

(ii) Find f(r) such that and f(1) = 0.

3. Find the directional derivative of = 4e2x – y+z at the point (1, 1, –1) in a

direction towards the point (–3,5,6).

4.Find the acute angle between the surfaces xy2z = 3x+z2 and 3x2-y2+2z at the

point (1,-2,1).5.

Find the value of a, b, c so that the directional derivative of

φ = axy2+byz+cz2x3 has a maximum of magnitude 64 in the direction of z-

axis.6.

Find the constants a and b so that the surface ax2 – byz = (a+2)x will be

orthogonal to the surface 4x2y + z3 = 4 at the point (1, –1,2).

7. Find the constants ‘a’ and ‘b’ such that the surfaces ax2-byz = (a+2)x and 4x2y+z3 = 4 may cut orthogonally at (1,-1,2).

8. Find the directional derivative of at along the unit normal to the

surface at In what direction, it is maximum? Find the maximum value.

Portions for first test:

Fourier series – complete.Complex variables - Cauchy -Riemann equations and their applications.Numerical analysis- complete.Vector analysis – Gradient.