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# 543 Review Sheet for M E 521 at PSU

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This 1 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Pennsylvania State University taught by a professor in Fall. Since its upload, it has received 30 views.

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Date Created: 02/06/15

Review of Complex Variables Author John M Cimbala Penn State University Latest revision 23 October 2007 Basic De nitions in the Complex Plane Define up of a real part x and an imaginary part iy The imaginary unity number i Complex number z is often represented graphically on the complex xy plane as shown in the sketch Com lex number z can also be written in terms of r and 6 where r is the magnitude of z zx jy re g where x and y are real numbers and z is a complex number made often thought of as a radius and 6 is the angle between the x axis and the ray z as shown on the sketch Mathematically 0 E a rcta rifyx x x The complex xy plane Some Rules and Review a b d Complex Conjugate The complex conjugate of a complex variable z x iy is obtained by changing the sign of the imaginary part of z zE x7zy requot9 Everywhere an 139 appears change it to a i Namely the complex conjugate of z is defined as Magnitude of a Complex Variable The magnitude also called the modulus of a complex variable z x iy is obtained by taking the square root of the V39Zzquot Note that this can be expanded to give rr product of z and its complex conjugate ie Miscellaneous Equations em e710 eig 7 e w Sln5 2 3921 19 r e cos 67151118 lg 19 26 6 7 7 1 7 s 2 j 7 1 2 cos 87 Forz1 7 rle and z2 7 rze zlz2 7 rlrze 8ng 39cos 8 7is39in19 Separating a Complex Function into Real and Imaginary Parts A trick to separate a complex function into its real and imaginary parts is to multiply and divide by the complex conjugate of the denominator of the function This always works because it guarantees that the denominator will become real Example Calculate the real and imaginary parts of F z cz where c is a real constant x iy ox Solution F c c 7 my xiy xiyx7iy x2y2 x2er2 So the real part is and the imaginary part is 763 x y x y Derivatives of Complex Functions Differentiation of complex functions is relatively simple because it follows the same basic rules as does differentiation of real functions product rule exponent rules etc Note F z is an analytic function in any region where a nite unique derivative dFdz can be de ned everywhere within the region lfFz is an analytic function then dFdz is independent of the orientation of dz in the xy plane Examples Fz z2 g 2z exponent rule z FZ 1112 E 1 natural log rule dz z FZ w E lnz 7i product rule combined with above rules z dz z z 22

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