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### M.sheela complex ppt

1. Introduction to Complex Variables Dr.M.Sheela Assistant Professor of Mathematics 1 ComPlex Analysis
2. Some Applications of Complex Variables 2  Phasor-domain analysis in physics and engineering  Laplace and Fourier transforms  Evaluation of integrals  Asymptotics (method of steepest descent)  Conformal Mapping (solution of Laplace’s equation)  Radiation physics (branch cuts, poles)
3. A complex number z may be thought of simply as an ordered pair of real numbers (x, y) with rules for addition, multiplication, etc.       2 2 1 ( ) 1, Re , Im cos sin arg ar , g tan i z x iy i j x z y z r i z re r z z r z x y x z z x y y                                   (from figure) (Euler formula (not yet proven!)) (angle notation) (angle notation) magnitude of argument or phase cos , sin z x r y r   of z x y  r Argand diagram (polar form) z 3 Note: In Euler's formula, the angle  must be in radians. Note: Usually we will use i to denote the square- root of -1. However, we will often switch to using j when we are doing an engineering example.
4.         1 2 1 1 2 2 1 2 1 2 z z x iy x iy x x i y y          Addition / subtraction: 4 Geometrically, this works the same way and adding and subtracting two- dimensional vectors:
5.               1 21 2 2 2 2 2 1 2 1 2 1 1 1 2 1 1 2 2 1 2 1 2 1 2 2 1 2 2 1 2 1 2 1 2 1 1 1 2 2 2 1 2 1 2 1 2 2 1 2 2 22 2 2 2 2 1 0 1 0 1 1 / / ( , ) ii i z z x iy x iy y x x y y i x y x y i i i z z re r e rr x iy x iy x x yx y e x iy z z x iy x x y y i y y x x y z z x y                                    64 7 48 Multiplication: Division:      1 21 2 2 2 1 2 2 1 1 2 1 2 2 2 2 / / ii i r z z re x r e e r y x x y                   5 Multiplication and division are easier in polar form
6. 6  We can multiply and divide complex numbers. We cannot divide two- dimensional vectors.  We can, however, multiply two-dimensional vectors in two different ways (dot product and cross product). Important points:
7.      * * * 1 2 1 2 1 2 2* 2 2 2 2 2 * / i i z x iy z z z z z z z z z r z x y z z re re           Note : Conjugation: Magnitude : z x y  r z  r z* 7
8. Euler’s Formula   2 3 0 2 3 0 2 4 3 5 0 1 2! 3! ! 1 2! 3! ! 1 ! 2! 4! 3! 5! cos sin n x n n z n n i n i x x x e x n x z x iy z z z e z n i e i n i e z                                                     L L L L Recall: Define extension to a complex variable ( ): (converges for all ) cos sin cos sin cos sin cos sin cos sin 2 2 i iz iz iz iz iz iz i e i e z i z e z i z e e e e z z i                      More generally, 8    cos cosh , sin sinh 2 2 2 z z z z z z e e e e e e iz z iz i i z i            
9. Application to Trigonometric Identities       2 2 22 2 2 cos2 sin 2 cos sin cos sin 2cos sin i i i e i e e i i                    W Many trigonometric identities follow from a simple application of Euler's formula : On the other hand, Equatingreal andimaginary parts of t              1 2 1 2 1 2 2 2 1 2 1 2 1 1 2 2 1 2 1 2 1 2 1 cos2 cos sin sin 2 2cos sin cos sin cos sin cos sin cos cos sin sin sin cos cos i i i i e i e e e i i i                                             W m he two expressions yields identities: On the other hand, two       2 1 2 1 2 1 2 1 2 1 2 1 2 sin cos cos cos sin sin sin sin cos cos sin                   m Equatingreal andimaginary parts yields : 9
10. DeMoivre’s Theorem 10  2 k  2 k  x y z z             2 2 cos sin cos 2 sin 2 co nn i n in n n i k i n knn n n z re r e r n i n n re r e r n kn i n kn k r                                 W W (DeMoivre's Theorem) Note that for aninteger, the result is of how is measured ( aninteger) independent  s sin n n i n z   
11. Roots of a Complex Number    cos sin nn i n in n th z re r e r n i n n n         W W (DeMoivre's Theorem) Applies also for not aninteger,but in this case, the result is independent of how is measu m no red. root ofp a cx om: pl t ea xE m l nue           22 22 6 32 1 1 1 1 1 31 3 2 2cos sin , 0,1,2, 1 2 2 8 8 2 2 cos sin , 0,1,2 6 3 6 3 2 cos 6 k n nii k ki ii i k n n n n n k k n n n nz re r e r i k n k k i e e i k                                                                    6 44 7 4 48 L roots ber : e.g.,    sin 2 cos 30 sin 30 2 6 i i                         3 2 1 2 i         3 , 2 2 2 cos sin 2 cos 90 sin 90 2 , 6 3 6 3 4 4 2 cos sin 2 cos 210 sin 210 3 , 6 3 6 3 i i i i i i i                                                                          11
12. Roots of a Complex Number (cont.)       1 3 222 1 1 1 1 3 , 8 2 , 3 k n k n n n ik z ii i n n n n n i i i i n z e n z re r e r e                      1 2 3 W "principal" throot of unityroot of Note that the throot of can also be expressedin terms of the :th root of unity  {   11 22 2 2 1 cos sin , 0,1, , 1n k nn ii k n k k e ie k n n n           1 2 3 L throot of unity where z x y 8i u v w   1/31/3 8w z i   Re Im 1 0  1 120  1 240  Cube root of unity 12 w u iv  Example (cont.)
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