9 Let g(z) = O is a z3 sin(2z)' a) a simple pole of g(z) b) a pole of order 2 of g(z) c) a pole of order 3 of g(z) d) a pole of order 4 of g(z) e) an essential singularity of g(z)
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- * = (-1), b = (² 2 Let. a P² = (3) and P a) Find Fourier Coefficient b) Find {a} such that ca. c) Find p*= {a,b} closest to p. of b with respect to a.a) Use Backward difference to find the Finite difference equation of the followi ди +a a'u xe (0,1) t20, b) Prove that the scheme is consistent?(b) Prove that the dual of c, is isometric to /'.
- Find the curl of = pz²cos($)a+zsin²(p)a None of the choices O2zsin(p) cos(p)a+2pz²sin(p)a+p²zcos(p)a₂ 2²_sin(4) ªz O 2zsin (4) cos(4) p² -a + 2zcos(p)a+ P O 2zsin(p) cos(p) −a+2pzcos(p)a+z²sin(p)a P O2zsin(4) cos(p)a+2p²zcos(p)a+pz²sin(p)a₂3. In cyone period fuuchion fit) i's defined as f(t) = e-t, -1s t s+l 3.1. Detezmine the complex Fourier Sevies expeu tevai - 3.2 . Delkrmi'ne if this feuenchion flt) satısty due Dirichlet's CundidicnS(5). If A, B and A+B are each non-singular and the products are conformable, prove that (a). A(A + B)-B = B(A + B)~'A %3D (b). A(A+ B)-1B (A-1+B-)-1
- a) Prove the one point rule - 3-version: (3x)(x = t ^ A) = A[x := t] if x is not free in t. b) Prove the dual result 6 on p.176. Prove (3x) AvBC = (3x) ÂС V (3x) BC. (Hint: Translate first to Standard notation) c) Prove using the auxiliary variable metatheorem: + (3x) (A → B) → (Vx)A → (3x)B1) Let z = 2+i be a point on a curve C. The direction of the tangent vector to C at z = 2+i is rotated through an angle θ under the action of the transformation f(z) =36z2+12z on C. Then the value of tan θ is. 2) The image of the line y = 5x under the mapping w = z2 is a straight line L. Then the slope of L is 3)A non-zero fixed point of the mapping w = 14z/(11z+2) is 4) If u(x,y) = 2x+3xy+c is the real part of an analytic function f(z), then the derivative of f at 5i, f'(5i) =1-b) [2]: Write the order and degree of the given ODE's. Check either these equations are linear or non-linear. Also check either the given equations are homogeneous or non-homogeneous. [CLO-1] a) y"+ x²y- 2y' = sinx, b) y"+ 4xy'-y+x = 0. x+1 c) x2y' + 3y 2x*e*. d) (y) -5(x – 1)y = 1. 1-c) 14]: Find the general solution of the given ordinary differential equations. [CLO-2|| a) y"+ 4y' +7y 0 b) y"+3y" + 3y' +y = 0
- Show that the path given by r(t) = (cos t,cos(2t), sint) intersects the xy-plane infinitely many times, but the underlying space curve intersects the xy-plane only twice.A periodic function f(x) = g(x+2x) f (x +27) us given by x+ π -TULX LO π-x, 0≤xs t 2 a) Sketch f(x) and state whether it is odd, even or her neither. b) Hence determine the Fourier serves of f(x). پالايهif A=xz3i - 2x²yzjt+2yz4k Find curlXA (Curl A) at point (1,-1,1)