Www.stpetershyd.com



SUBJECT: MATHEMATICS-4UNIT-1SHORT QUESTIONS:Test whether ez is analytic or notFind the singular point of tanzWrite the C-R equations in the cartesian formWrite the C-R equations in polar formDefine Harmonic function.Show that f(z) = z+2z is not analyticFind the value of k so that f(z) is harmonic where f(z)=x2+2x+ky2Define singular point .Determine whether the function 2xy+i(x2-y2)is analytic.Define regular function.ESSAY TYPE QUESTIONSDetermine p such that the function f(z) = 1 2log(x2+y2)+tan-1pxy is an Analytic functionIf u(x,y)and v(x,y) are Harmonic functions in the region R, prove that the function(?u?y-?v?x)+i(?u?x+?v?y) is an Analytic functionS.t u(x,y)=e2x(x cos2y-y sin2y) is Harmonic and find its Harmonic conjugateFind the Analytic function f(z)= u(x,y)+i v(x,y) if u-v=cosx+sinx-e-y2 cosx-ey-e-y and f(π2)=0.Prove that u=(e-x(x2-y2)cosy+2xy siny) is harmonic and find the analytic function whose real part is u.Find the imaginary part of an analytic function whose real part is ex(x cosy-y siny)Prove that u=e-2xysin(x2-y2)is harmonic find the conjugate function and express u+iv as analytic functionFind the analytic function whose real part is u(r,?)= -r3sin3?If the potential function is log(x2+y2) find the flux function and the complex potential functionShow that the real and imaginary parts of an analytic function are harmonicUnit –IISHORT QUESTIONS:1. Define line integral.2. Evaluate x+ydx+x2ydy along y=3x between (0,0)and (3,a)3. Evaluate 01+iz2dz along y=x24. State Cauchy’s theorem.5. State Cauchy’s integral formula 6. State Taylor theorem.7. State Laurent theorem8.Define removable singularity9. Define essential singularity10. Define isolated singularityESSAY TYPE QUESTIONSEvaluate 01+ix2+iydz along the path y=x and y=x2Evaluate (z+1)dz=0 where c is the boundary of the square whose vertices at the points z=0,z=1,z=1+i and z=iVerify cauchy’s theorem for the function f(z)=3z2+iz-4 if c is the square with the vertices 1±i,-1±iEvaluate (z2+3z+2)dz where c is the arc of the Cycloid x=a(?+sin?),y=a(1-cos?) between the points(0,0) and (aπ,2a)Evaluate sin6zz-π63dz around the circle c:z=1Evaluate logzz-13dz where c:z-1=12using cauchy’s integral formulaObtain the Taylor series expansion of f(z)=ezz(z+1) about z=2Expand f(z)=sin z as a Taylor series about z=π2Find the Laurent series expansion of f(z)=1z2(1-z) and find the region of convergenceFind the Laurent series expansion of the function f(z)=z2-1z+2(z+3) if 2<zFind the residue of f(z)=z3z-14z-2(z-3) at z=1Find the poles and residues of f(z)=z+1z2(z-2)Evaluate z-3z2+2z+5 dz where c is the circle given by z+1+i=2Evaluate 02πd?(5-3 sin?)2 using residue theoremUNIT-IIISHORT QUESTIONS:1. Define bilinear transformation2. Define cross ratio property of bilinear transformation3. Fine the bilinear transformation that maps z=-2,0,2 into w= 0,i,-i4. Find the mobius transformation that maps 0,1,∞ into -5,-1,3 find fixed points5. .Find the mobius transformation that maps 0,1,∞ into i,1,-i6. Show that w =z-1z+1 maps unit circle in w-plane onto imaginary axis in z-plane.ESSAY TYPE QUESTIONSBy contour integration evaluate 0∝dx1+x2Evaluate 0∝logx dx1+x2Find the Bilinear transformation that maps the points z1=1-2i, z2=2+i, z3=2+3i in to the points w1=2+2i, w2=1+3i and w3=4 respectively. Find the fixed and critical pointsFind the Bilinear transformation that maps the points z1=1, z2=i, z3=-1 in to the points w1=2, w2=i and w3=-2 respectively. Find the fixed and critical pointsShow that the transformation w=4z transforms the straight line x=c in the z-plane into a circle in the w-planeUnder the transformation w=z-i 1-iz find the image of the circle (i)w=1(ii)iz=1 n the w-planeShow that under the transformation w=1z , a circle x2+y2-6x=0 is transformed into a straight line in the w-planeShow that the condition for transformation w=az+bcz+d to make the circle w=1 correspond to a straight line in the z-plane is a=cDiscuss the transformation w=z2Define conformal transformation and find the invariant points of the transformation w=z-1z+1.Find the Bilinear transformation that maps the points 1+i,-i,2-i of the z-plane into the points 0,1,i respectively of the w-plane. Find the fixed and critical points.Find the image of z=2 under the transformation w=3z .UNIT –IVSHORT ANSWER QUESTIONSDefine periodic function and write examples .Define even and odd function.Express the function f(x) as the sum of an even function and an odd functions.Find the functions are even or odd (i) x sinx+cosx+x2coshxDefine Euler’s formulaeWrite Dirichlet’s conditions.State Fourier integral theorem.Write about Fourier sine and cosine integralWrite the properties of Fourier transformFind the finite Fourier sine transform of f(x)=2x in (0, 2 )ESSAY TYPE QUESTIONSObtain the Fourier series expansion of f(x) given that fx=(π-x)2 in 0<x<2π and deduce the value of112+122+132+142±-----=π26Obtain Fourier cosine series for f(x)=x sin x ,0< x < π show that 11.3-13.5+15.7-17.9+---=π-24Find the Fourier Series to represent the function f(x)=sinx in –π< x< πFind the Fourier series to represent f(x)=x2 in (0,2π).If f(x)= sin x expand f(x) as cosine series 0< x < πExpand the function f (x) =x2 -2 as a Fourier series in (-2,2) Find cosine and sine series for fx=π-x in(0,π)Find the finite Fourier sine and cosine transforms of f(x) = sinax in (0,π).UNIT-VShort answer questions.1. Classify the differential equation 2 uxx+uxy-uyy+ux+3uy=0.2. Classify the differential equation uxx- y4uyy=2y3uy3. Classify the differential equation 3 uxx+4uxy- 6uyy=04. Solve ?u?x = 2?u?t +u, u(x,0) = 6 e -3x using method of separable.5.Solve 2 ?2z?x2 -?z?y=0 by the product method6. Solve 4 ?u?x +?u?y =3u given that u=4e-3x when t=07. Solve ?u?x +4=?u?t if u=4e-3x when t=0.ESSAY TYPE QUESTIONS1.The ends of a uniform string of length 2l are fixed .The initial displacement is y(x,0)=3x(2l-x).0<x<2l,while the initial velocity is zero. Find the displacement at any x from the end x=0 at any time ‘t’.2.Find the displacement of the string of length l vibrating between fixed end points with initial displacement given by Y(x,0)=2pxl , 0<x<l22p(l-x)l , l2<x<l3.A tightly stretched string with fixed ends at x=0 and x=l is initially in a position given by y(x,0)=u0 Sin3(πxl).If it is released from rest from this position,find the displacement ‘y’ at any distance ‘x’ from one end and any time ‘t’.4.A tight string of length 50 cm fastened at both ends,is disturbed from its position of equilibrium by implanting to each of its points an initial velocity of magnitude 1 cm for 0<x<50.Find the diaplacement function y(x,t).5.Find the solution of the one dimensional wave equation ?2 y?t2=4?2 y?x2 subject to the conditions y(0,t)=0,y(5,t)=0, y(x,0)=0,?y(x,0)?t =3 sin 2πx- 2 sin 5πx6.Solve the problem of the vibrating string for the following conditionsY(0,t),y(l,t) ?y?t(x,0) = x (x-l),o<x<lY(x,0) =x:0<x<l2l-x:l2<x<l7.A uniform rod of length 50 cm with insulated sides in initially at a uniform temperature 100 0c its ends are kept at 00c.find the temperature distribution8.Solve ?u?t =α2?2u?x2subject to the conditions(i) u is not infinite for t →∞(ii) u=0 for x=0 and x =π for all t(iii)u = πx-x 2 for t =0 in (0, π)9.The ends A&B of a rod 40 cm long their temperature kept at 00c and 80 0c respectively until steady state conditions prevail. the temperature of the end B is then suddenly reduced at 400cand kept 80,while that of the end A is kept at 00c.Find the temperature u(x,t) at any instant of t.10.A rod of length l has it’s end A and B kept at 00c and 1500crespectively until steady state condition prevail. If the temperature at b is reduced to 00c and kept 80 while that of A is maintained so, find the temperature u(x,t) at a distance from A and at time t. ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download