... , C2 and C2 C z dz = z3 − C1 z− + √ C2 z− C3 z− + = 2 + 2 + 2 z− √ √ √ z− z− z √ dz e 2 /3 z − e− 2 /3 z √ √ dz 2 /3 z − 9e z − e− 2 /3 z √ √ dz z − e 2 /3 z − e− 2 /3 z− 9 √ z e 2 /3 ... C2 -4 -6 C C1 -2 C3 -2 -4 Figure 11 .4: The contours for z z −9 the Cauchy Integral Formula to evaluate the integral C d sin z sin z dz = 2 − 4) dz z − z=0 cos z sin z = 2 − z − (z − 4) 2 ... continuously deformed to C2 on the domain where the integrand is analytic Thus the integrals have the same value 5 14 -4 -2 -2 -4 Figure 11 .2: The contours and the singularities of 3z +1 -6 -4 -2 -2 -4 -6...
... figuring out budgets are all good practice for life after graduation -> grow into adulthood - A better perspective on what is going on around them 22 You have been told that dormitory rooms at ... that going to classes should be optional for students Which point of view you agree with? Use specific reasons and details to explain your answer TOEFL 14 - Why should attend the class? - Personal ... to absorb information and apply what they’ve learned to other situations - Teach them how to work with the other members of the class - Teach students responsibility and discipline 20 Some people...
... necessarily the best format for dissemination of documents Formats for electronic documents and images - Portable document format (pdf) – page When to use PDF? Portable Document Format (PDF) is a ... format to use for its dissemination… The best format for document delivery depends on what you’re doing with the document To create and edit a document, Microsoft Word can be an appropriate format ... objects Formats for electronic documents and images - Portable document format (pdf) – page Summary • PDF (Portable Document Format ) is a procedural mark-up language that allows page-formatted...
... − sin2 x = lim x→0 24 cos2 x − 8x2 cos2 x − 64x cos x sin x − 24 sin2 x + 8x2 sin2 x = = lim It is easier to use a Taylor series expansion lim csc2 x − x→0 x2 x2 − sin2 x x→0 x2 sin2 x x2 − (x ... x→0 2x cos x sin x + 2x sin2 x − cos2 x + sin2 x = lim x→0 2x cos2 x + 8x cos x sin x + sin2 x − 2x2 sin2 x cos x sin x = lim x→0 12x cos2 x + 12 cos x sin x − 8x2 cos x sin x − 12x sin2 x cos2 ... , (2y(x) − x)3 105 Solution 3.15 a f (x) = ( 12 − 2x )2 + 2x( 12 − 2x)( 2) = 4( x − 6 )2 + 8x(x − 6) = 12( x − 2) (x − 6) There are critical points at x = and x = f (x) = 12( x − 2) + 12( x − 6) = 24 ( x...
... are ux = vy and uy = −vx −(x − 1) + y −(x − 1 )2 + y 2( x − 1)y 2( x − 1)y = and = ((x − 1 )2 + y )2 ((x − 1 )2 + y )2 ((x − 1 )2 + y )2 ((x − 1 )2 + y )2 2 The Cauchy-Riemann equations are each identities ... satisfied for z = Note that the form fx = −ıfy will be far more convenient than the form ux = v y , uy = −vx 40 4for this problem fx = 4( x + ıy)−5 e−(x+ıy) −ıfy = − 4( x + ıy)−5 ı e−(x+ıy) 44 = 4( x ... lim = lim z→0 cos (z ) =1 379 − 4z sin (z ) lim z→(nπ)1 /2 z − (nπ)1 /2 = lim 2) 1 /2 2z cos (z ) sin (z z→(nπ) = 1 /2 (−1)n 2( nπ) Example 8 .4. 4 e1/z is singular at z = The function is not analytic...
... function Recall that for analytic functions f , f = fx = −ıfy So that fx + ıfy = 22 + = ∂u2 ∂v dw dz −1 dw dz 22 dw + = 2 ∂u ∂v dz −1 222 + ∂x2 ∂y 22 + ∂x2 ∂y Solution 8.16 ... analytic 42 3 We calculate the first partial derivatives of u and v u x = ex −y uy = 2 e (x cos(2xy) − y sin(2xy)) x2 −y (y cos(2xy) + x sin(2xy)) vx = e x2 −y (y cos(2xy) + x sin(2xy)) vy = e x2 −y ... velocity field v= v= φ 2x(x + y − 1) 2y(x2 + y + 1) ˆ ˆ x+ y x4 + 2x2 (y − 1) + (y + 1 )2 x + 2x2 (y − 1) + (y + 1 )2 43 1 Figure 8.9: Velocity field and velocity direction field for φ = ln r − θ The...