Read e-book online Hyperlearning MCAT Science Workbook PDF

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This package deal contains the Hyperlearning MCAT technology Workbook, in addition to the Verbal Workbook and in-class compendium, all in excellent .

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M. (z) , z-a '¥ (19) where f/J (z) is analytic near and at a. The coefficient a_I is called residue of f (z) relative to the pole a. Consider now the integral ¢ f (z) dz over a circle °whose center is at a and whose radius R is assumed to be small enough so that f/J (z) is analytic inside and on the circle. We have: f I m ¢(z~a)r f f(z) dz = a- r + f/J (z) dz . (20) a r=1 a a The last integral on the right is zero [see Eq. (12)] and the contour integrals on the right vanish unlessl r = 1. Butforr = 1, we have (putting z-a = ee j Il): 1= 2" ~f(z)dz=a_l ~~=a-l ~ ejej~d() =2nja-l.

8 = VA2 B + J32 = . 5. Symbolic Method for Solving Linear Differential Equations given by (3 B tanrp= ~=A. (44) Equation (41) can thus be written in the form 8 VA B2 [cos rp cos w t + sin rp sin w t] = VA + B2 cos (w t - rp) . (t) = 2 (45) 2 If we had identified (J,. with sin rp, (3 with cos rp, we would have obtained similar result in terms of the sine: 8 where (t) = VA2 + B2 sin (wt + rp), [1 (46) (47) We shall usually select the real part of a complex solution as the real solution and, consequently, shall always use the cosine, Eq.

3), one can be interpreted as a potential, . au. au ~x = - ax' ~y = - ay , (8) the other as the stream function, . av ~x = ay , . av ax ~y=- ~, (9) for incompressible flow. , and the quantities ~x, ~y represent the velocity components of the flow field in the x and y directions, respectively. Equations (8) and (9) show that the real part of an analytic function can always be interpreted as a potential, the imaginary part as a stream function. In the theory offunctions, the infinite plane - 00 ~ x ~ 00, - 00 ~ y ~ 00 is imaged on the surface of a sphere.

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