North MaharashtraUniversity ; Jalgaon.

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1 North MaharashtraUniversity ; Jalgaon. Question Bank S.Y.B.Sc. Mathematics (Sem II) MTH. Functions of a omplex Variable. Authors ; Prof. M.D.Suryawanshi (o-ordinator) Head, Department of Mathematics, S.S.V.P.S. s L.K.Dr.P.R.Ghogrey Science ollege, Dhule. Prof. U.S.Shirole Department of Mathematics, S.S.V.P.S. s L.K.Dr.P.R.Ghogrey Science ollege, Dhule. Prof. A.G.Patil Kisan Mahavidyalaya, Parola, Dist.- Jalgaon. Prof. J.G.Patil S.S.V.P.S. s Arts, ommerce and Science ollege, Sindkheda Dist. Dhule.

2 Unit : Functions of a omplex Variable. I) Questions of Two marks : ) The lim [ x + i( x y )] is i ) Does lim exist ) What are the points of discontinuties of f() = + where = x+iy. ) Write the real and imaginary parts of f() = 5) Find the limit, lim [ x + i( x + y )] i ) Does ontinuity at a point imply differentiability there at. Justify by an example. ) Define an analytic function. ) Define singular points of an analytic function f(). ) Find the singular points for the function f() = ( + )( 5) Define a Laplace s Didifferential Equation for Φ ( x, y ). 6) What is harmonic function? 7) What do you mean by f() is differentiable at? 8) Is the function u =.log ( y ) x + harmonic? 9) When do you say f() tends to a limit as tends to? ) State auchy- Riemann equations. ) State the necessary condition for the function f(x) to be abalytic. ) Every differential function is continuous. True or False. II) Multiple hoice Questions : ) If lim [x+i(x+y)] = p+iq, then (p,q) = i (i) (,) (ii) ( -,) (iii) (,-) (iv) (-,-) xy ) The function f() = when and f() = is x + y (i) ontinuous at =, (ii) Discontinuous at = (iii) Not predictable, (iv) onstant )A ontinous function is ifferential : (i) True,(ii) False. (iii)true & False, (v) True or False ) A function Φ ( x, y ) satisfying Laplace equation is called (i) Analytic (ii) Holomorphic (iii) Harmonic, (iv)non-hormonic ) Afunction f() = e is (i) Analytic everywhere, (ii) Analytic nowhere (iii) only differentiable, (iv) None + )

3 ) If f() = u iv is analytic in the -plane, then the -R equations satisfied by it s real and imaginary parts are, (i) u x = u y ; u y = vx (ii) u x = v y,u y = vx 7) An analytic function with constant modulus is (a) onstant, (b) not constant, (c) analytic, (d) None of these. 8) A Milne Thomson method is used to construct a) analytic function, b) ontinuous function c) differentiable function, d) None of these. III) Questions for Four marks ; ) Define he continuity of f() at = and examine for continuity at = the function x y( iy x ) f() = ; 8 ( x + y ) ) Define limit of a function f(z). + 8 Evaluate ; lim Πi ) Prove that a differentiable function is always continuous. Is the converse true? Justify by an example. ) Use the definition of limit to prove that, lim [ x + i( x + y )] = + i i 5) Show that if lim f() exists, it is unique 6) Prove that lim does not exist x y( y ix ) ) Prove that lim 6 ( x + y ). does not exist, where 5) Evaluate : lim 5 i + i i 6) 9) Evaluate : lim + i + i ) Examine for continuity the function, f() = + at i i = +i ; =i, at = i )If f() = when i and f(i) = +i, examine f() for continuity i At =i.. )Show that the function f() = is continuous everywhere but not differentiable. ) Define an analytic function. Give two examples of an analytic function. ) Show that f() = is not analytic at any point in the -plane.

4 5) State and prove the necessary condition for the f() to be analytic. Are these conditions sufficient? 6) State and prove the sufficient conditions for the function f() to be analytic. 7) Prove that a necessary condition for a complex function w = f() = u(x,y)+iv(x,y) to be analytic at a point =x+iy of its domain D is that at (x,y) the first order partial derivatives of u and v with respect to x and y exist and satisfy the auchy Riemann equations : u x = v y and u y = v x. Prove that for the function F() = U(x,y) + V(x, y), if the four partial derivatives U x, U y, V x and 8) V y exist and are continuous at a point = x + iy in the domain D and that they satisfy auchy- Riemann equartions: U x = V y ; U y = -V x at (x, y), then F() is analytic at the point = x + iy. 9) Show that the function defined by F() = Ιxy Ι, when and F() =, is not analytic at = even though the -R equations are satisfied at =. Define F() = 5 I I - ; if ) = ; if =. Show that F() is not analytic at the origin even though it satisfies -R equations at the origin. x ( + i) ) y ( i) ) Show that the function F() = when and F() = is continuous at x + y = and -R equations are satisfied at the origin. ) If F() and F () are analytic functions of, show that F() is a constant function. ) If F() is an analytic function with constant modulus, then prove that F() is a constant function. ) Show that + =. x y 5) Show that F() = e is not analytic for any. 6) x y Show that lim does not exist. x + y 7) dw Show that if W = F() = x - iy, then does not exist. d x ( + i) ) y ( i) 8) Show that the function F() = when and F() = is continuous and x + y that -R equations are satisfied at the origin but F () does not exist. xy ( x + iy) Show that the function defined by F() = ; 9) x + y = ; =. satisfies.r. equations at = but not analytic there at. If F() is an analytic function of then show that ) (i) x x + y F ( ) = F ( ) (ii) + [ RF )] y ( = F ( ) UNIT : Laplace Equation and omplex Integration

5 I Questions of TWO marks ) Define Laplace Differential equation. ) Define harmonic and conjugate harmonic functions. ) True or False: i ) If F() is an analytic function of, then F() depends on. ii) If F() and F () are analytic functions of, then F() is a constant. iii) An analytic function with constant modulus is constant. ) Is u = x - y a harmonic function? Justify. 5) Show that v(x, y) = x - y + x is harmonic function. 6) Show that u(x, y) = e -y sinx is a harmonic function. 7) Prove or disprove: u = y x y is a harmonic function. 8) Show that v = x - xy satisfies Laplace s differential equation. 9) State auchy-goursat Theorem. ) Define simple closed curve. ) Define the term Simply connected region. ) Define Jordan urve. ) State Jordan urve theorem. ) Evaluate d where is circle a =. a 5) +i Evaluate d along the line x = y. ) ) ) ) 5) II Multiple hoice questions mark each The harmonic conjugate of e x cosy is. (a) e x cosy + c (b) e x siny + c (c) e x + c (d) None of these The harmonic conjugate of e -y sinx is. (a) e -y cosx + c (b) e -y sinx + c (c) e -x cosy (d) None of these The value of the integral ( - /) d where is the curve y = x - x + x joining points (,) and (,) is given by. (a) i (b) i (c) 5 (d) None of these The value of e d will be (a) e (b) / (e +) (c) / (e -) (d) None of these The value of the integral of / along a semicircular arc from - to in the clockwise direction will be. (a) ero (b) πi (c) πi (d) None of these Questions of THREE marks

6 ) If F() = u + iv is an analytic function then show that u and v both satisfy Laplace s differential equation. ) If F() = u(x,y) + iv(x, y) is an analytic function, show that F() is independent of. ) Explain the Milne-Thomson s method to construct an analytic function F() = u + iv when the real part u is given. ) Explain the Milne-Thomson s method to construct an analytic function F() = u + iv when the imaginary part v is given. Find an analytic function F() = u + iv and express it in terms of if 5) u = x - xy + x - y +. 6) Find an analytic function F() = u + iv if, v = e -y sinx and F() =. 7) Find an analytic function F() = u + iv where the real part is e -x sin (x y ). 8) y cos x + sin x e If F() = u + iv is analytic function of = x + iy and u - v =, find F() if y y cos x e e f(π / ) =. 9) Show that the function F() = e -y sinx is harmonic and find its harmonic conjugate. ) Use Milne-Thomson s method to construct an analytic function F() = u + iv where u = e x ( xcosy - ysiny). ) Use Milne-Thomson s method to construct an analytic function F() = u + iv where v = tan - (y / x). ) Determine the analytic function F() = u + iv if u = x - y and F() =. ) Find by Milne-Thomson s method the an analytic function F() = u + iv where v = e x ( xsiny + ycosy). ) If u = x - y y and v = x + y, then show that u and v satisfy Laplace equation but u + iv is not an analytic function of. 5) Show that if the harmonic functions u and v satisfy.r. equations, then u + iv is an analytic function. b 6) If F() is analytic in a simply connected region R then F() d is independent of the path a of the integration in R joing the points a and b. 7) Evaluate d where is the arc of the parabola y = ax (a >) in the first quadrant from the vertex to the end point of its latus rectum. 8) Evaluate d where is circle a =. a 9) Evaluate ) Evaluate ( y x x i) d where is the straight line joining to + i. ( y x x i ) d where is the straight line joining to i first and then i to + i. ) Show that the integral of / along a semicircular are from - to has the value πi or πi according as the arc lies below or above the real axis. 5

7 Show that if F() is an analytic function in a region bounded by two simple closed curves ) and and also on and, then F() d = F() d. ) State auchy s theorem for integrals and verify it for F() = + rounder the contour =. ) If is a circle a = r, prove that ( a ) n d = ; n being an integer other than -. 5) d Evaluate where is the circle with centre at origin and radius a. 6) Verify auchy-goursat Theorem for F() = + taken round the unit circle =. 7) Verify auchy s integral Theorem for F() = round the circle =. 8) Verify auchy s Theorem for F() = around a closed curve. where c is the rectangle bounded by the lines : x =, x =, y =, y=, Use auchy-goursat Theorem to obtain the value of e d, where is the circle = and 9) π cosθ cosθ deduce that (i) e sin( θ + sinθ ) = (ii) e cos( θ + sinθ ) =. UNIT : auchy s Integral Formulae and Residues. I Questions of TWO marks ) State auchy s integral formula for F(a). ) State auchy s integral formula for F (a). ) Evaluate by auchy s integral formula ) Evaluate ( e = ) d. 5) e Evaluate ( ) 6) Evaluate by auchy s integral formula + d π d where is the circle =. 7) Evaluate by auchy s integral formula + 8) Evaluate d = 9) Define apower series. ) State Taylor s series for F() about = a. ) State Laurent s series for F() about = a. ) Expand in Taylor s series: e ( d where is the circle =. where is the circle =. d where is the circle =. ) where is the circle = /. Use auchy s integral formula. for <. 6

8 ) Expand in Laurent s series: F() = valid for <. ) Define ero of an analytic function. 5) Define singular point of an analytic function. 6) State the types of singularities. 7) Define a pole of an analytic function. ) ) ) ) 5) A power series R = (a) n= n II Multiple hoice Questions mark each a ( a) n converges if a <R ( b) a >R (c) a =R (d) None of these If F() is an analytic function at = a, then it has a power series expansion about = a. (a) Statement is true ( b) Statement is false (d) None of these The region of validity for Taylor s series about = of the function e is (a) = ( b) < (c) < (d) > The region of validity of for its Taylor s series expansion about = is + (a) < ( b) > (c) = (d) None of these The expansion of is valid for (a) < ( b) < (c) > (d) None of these 6) sin If F() =, then = is its. (a) Removable singularity ( b) Isolated singularity (c) Essential singularity (d) None of these 7) Z = is a.. of F() = ( ). (a) ero ( b) simple pole (c) double pole (d) None of these 8) The residue of F() = + at a pole of order is (a) ( b) - (c) (d) None of these 9) The singular points of F() = ( ) are.. (a),, - ( b),, (c), - (d) None of these III Questions of FOUR marks ) State and prove auchy s integral formula for F(a). ) State and prove auchy s integral formula for F (a). ) + Evaluate by auchy s integral formula d, where is + i) the circle = ii) the circle = ½. 7

9 ) sinπ + cosπ Evaluate + where is the circle = + Use auchy s integral formula to evaluate 5) d, where is the boundary of a square with vertices + i, - + i, - - i and i traversed counter clock wise. 6) State auchy s integral formula for F n e (a) and use it to evaluate d. ( ) 7) Evaluate = π e d. And hence deduce cosθ cosθ i) e cos(sinθ ) = π and ii) e sin(sinθ ) = π 8 = 8) State Taylor s series for F() about = a and find the Taylor s series expansion of F() = sin in powers of. 9) sinπ Evaluate d.expand in Taylors series ( ) ) = Expand in Taylor s series: for <. ) Expand in Taylor s series about =, the functions F() = and g() = cosh. ) Expand in Taylor s series about = the following functions: (i) sin,(ii) sinh, (iii) cos. ) Expand F(x) = e in Taylor s series expansion about =. State the region of its validity. ) π Expand sin in powers of ( - ). 5) Show that 5 7 tan = ; < ) Expand in Taylor s series: F() = for <. ( )( ) 7) Expand F() = for < in Taylor s series. 8) Expand F() = in Laurent s series valid for <. 9) Expand F() = ( ) in powers of for (i) < (ii) < < and (iii) >. ) + 5 Expand on the annulus < <. ( )( + )

10 ) Prove that = = n o n n+ where < <. ) Find poles and residues at these poles of f() = residues.. ( ) also find the sum of these ) Find the sum of residues at poles of f() = + a e ) Find the residues of f() = ( )( )( ) at its poles. ) Find the residues of ( ) + at = i. 5) ompute residues at double poles of f() = + + ( i ). ( + ) 6) Use auchy s integral formulae to evaluate ( Any one ) i) d, where is the circle Z = ( + )( + ) d ii), Where is the circle =..( + ) iii) = e ( ) d. d iv) ( + ), Where is the circle i =. 7) Show that e ( + ) 8Πie d =, where is the circle =. 8) Expand f() = ( + )( + ) in the regions: i), ii), iii) ) Expand : for + i), ii) and iii) Unit - auchy s Residue Theorem and ontour Integration I) Questions of Two Marks ; 9

11 ) State auchy s Residue Theorem. + ) find all poles of f() = ( )( + 9) ) Find the residues of f() a =, Where, f() = e. ( ) ) Define a rational function. + 5) Find the residues of f() =. ( )( + 9) 6) Find Zeros and poles of f() = e ( ) 7) Find all eros and poles of f() = ( + ) ( + ) 8) lassify the poles of f() = 9) Which of the poles of f() = Lies inside the circle =. 7) Which of the poles of f() = 8) Find the poles of f() = plane. ( + ) ( + )( + ) ( lies in the upper half of the - plane. + + a )( + b which lie in the lower half of the complex ) 9) Find all eros and poles of f() = ( + )( and lassify them. + ) ) Find all eros and poles of ) Find all eros and poles of cos x x + ( x x.sin x + a )( x + b ) III) Questions of Six Marks : ) State and prove auchy s Residue Theorem. 5 ) Evaluate by auchy Residue Theorem : d, where is the.( )

12 ircle = taken ounter clockwise. )Evaluate: + by auchy s Residue Theorem, where is ( )( ) d + 9 i) the circle =, ii) the circle = )Evaluate : e d, where is the circle = traversed in positive direction, ( ) 5) Evaluate : d by auchy s Residue Theorem, where is the ( + ) ( + ) rectangle formed by the lines x= +, y = +. 6) Use auchy s residue theorem to evaluate 7) Use ontour integration to evaluate 5 + cosθ Π 8) Evaluate : 5 + sinθ Π 9) Evaluate : (cosθ + ) Π Π ) Use method of contour integration to evaluate dx ) Apply calculus of residues to evaluate x + + d =.( + ) x x + ) Evaluate by contour integration dx x + x 9 + sin θ ) Evaluate : dx ( x + a )( x + b ) ; where a, b x ) By ontour integration, evaluate dx ( x + )( x ) + cos x 5) Evaluate : dx by using ontour integration. x +

13 x.sin x 6) Evaluate by contour integration, dx ( x + a )( x b ) + where a, b. e 7) Evaluate by auchy s residues theorem d Π = 8) Evaluate by contour integration 5 + sinθ Π 9) Evaluate by ontour integration + cosθ Π ) Evaluate + cosθ + sinθ Π Π cos ) Evaluate : θ 5 + cosθ d ) Evaluate : ) Evaluate, ) Evaluate : 5) Evaluate ; 6) Evaluate : dx x x + x dx + x dx ( x + ) ( x cos x x + dx + dx 6 x + 5 ) x sin x 7) Evaluate : dx x a 8) Use ontour integration to prove that = Π + sin θ Π Π 9) Show that Π Π = = a + b cos θ a + b sinθ a Π b where a, b cos mx ) Prove that dx = Π e ma, m x + a a and a

14 x.sin ax Π a ) Prove that, dx = e sin a; a x + I) Multipal hoice Questions; ) The poles of f() = are a a) + i,b),, c) + ai, d) None of these. e ) The poles of f() = ( ) + are a) + i, b),,c) + i,d) None of these. ) The sum of the residues at poles of f() = + a a) sin a, a b), c), d) None of these. e is ) The sum of the residues of f() = ( ) + is a), b), c) -,d) None of these. + 5) The residue of f() = at = is a), b), c ) -, d ) None of these. 6) The sum of residues at its poles of f() = a), b), c ) -, d ) None of these. ( ) is ) The simple poles of f() = are a), b)-, c) -,- d) None of these. 8) For the function f() =. + ( + ) a), b), c), d) None of these., the pole = has order