UNIT- III ANALYTIC FUNCTIONS

PART - B

IMPORTANT QUESTIONS.

1. If f (z) is a regular function of z, prove that ( ∂2

∂ x2 +∂2

∂ y2 )|f (z )|2 = 4¿ f ' ( z )∨¿2¿.

2. If f(z) = u+iv is a regular function of z, then show that ( ∂2

∂ x2 +∂2

∂ y2 )|f (z )|p = p2∨f ( z )∨¿p−2 ¿

¿ f ' ( z )∨¿2¿.

3. If f (z) = u+iv is analytic, prove that ( ∂2

∂ x2 +∂2

∂ y2 )log|f ' ( z )∨¿0.

4. The real and imaginary parts of an analytic function W=u(x,y)+iv(x,y), satisfies the Laplace equations ie.,∇2u=0.5. If W=u(x,y)+iv(x,y) analytic function, the curves of family u(x,y)=c1 and v(x,y)=c2 cuts orthogonally where c1 and c2 are constants.6. Find the analytic function f(z)=u+iv whose real part is

(i) u = ex(xcosy-ysiny)

(ii) u = e− x[(x2− y2 ¿cosy+2 xysiny ¿

(iii) u = sin 2 x

cosh2 y−cos2 x and u+v =

sin 2 xcosh2 y−cos2 x

.

7. Find the analytic function f(z)=u+iv whose imaginary part is

(i) v =e2x ( ycos2 y+xsin2 y ).

(ii) v = e− x ( xcosy+ ysiny )

(iii) v = ex2− y2

sin (2 xy ).

8. Determine the analytic function of u+v = ex (cosy+siny )∧¿2u+v = ex (cosy−siny ).

9. Prove that the function u = x3−3 x y2+3 x2−3 y2+1is harmonic. Find the conjugate

harmonic function and corresponding f(z).

10. Find the bilinear transformation that maps,

(i) z = 1,i,-1 & w = i,0,-i

(ii) z = -2,0,2 & w = 0,i,-i

(iii) z= ∞,I,0 & w = 0,I,∞.

(iv) z = 0,1,∞ & w = -5,-1,3

11.Find the image of |z-2i|=2 under the transformation w = 1/z.

12.Find the image of the strips (i)1/4<y<1/2 (ii) 0<y<1/2 under the transf., w = 1/z.

13.Discuss the transformation w = 1/z.

PART-A

IMPORTANT QUESTION

1.Define analytic function?

2.State sufficient conditions of C-R equation to be analytic?

3.Write down the polar form of C-R equation?

4.Test whether the function w =2xy +i(x2− y2 ¿ is analytic.

5. Find a,b,c if f(z)=(x-2ay)+i(bx-cy) is analytic.

6.Check f(z)=z3 is analytic & find dw/dz.

7.Verify whether f(z)= sinhz is analytic using C-R equ.,

8. Check f(z)=ez is analytic & find dw/dz.

9.Test the analytic function of f(z)=zn and find its derivatives.

10.Check it f(z)= |z2∨¿ is analytic or not.

11.S.T the function with constant real part is constant.

12.S.T an analytic function with constant modulus is constant.

13.Define harmonic function?

14.P.T u = ex cosyis a harmonic function.

15.find the conjugate harmonic of u = 12

log (x2+ y2 ) .

16.Define bilinear transformation?

17.Find the invariant point of z-1/z+1.

18.Define conformal mapping?

19.Find the critical point of (i) w = z +1/z, (ii) w = sinz.

20.Find the image of the circle |z|=α under the transf., w = 5z.

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