tailieunhanh - Advanced Engineering Math II Math 144

This clear, pedagogically rich book develops a strong understanding of the mathematical principles and practices that today's engineers need to know. Equally as effective as either a textbook or reference manual, it approaches mathematical concepts from an engineering perspective, making physical applications more vivid and substantial. Its comprehensive instructional framework supports a conversational, down-to-earth narrative style, offering easy accessibility and frequent opportunities for application and reinforcement | Advanced Engineering Math II Math 144 Lecture Notes by Stefan Waner First printing 2003 Department of Mathematics Hofstra University Simpo PDF Merge and Split Unregistered Version - http 1. Algebra and Geometry of Complex Numbers based on of Zill Definition A complex number has the form z x y where x and y are real numbers. x is referred to as the real part of z and y is referred to as the imaginary part of z. We write Re z x Im z y. Denote the set of complex numbers by C . Think of the set of real numbers as a subset of C by writing the real number x as x 0 . The complex number 0 1 is called i. Examples 3 3 0 0 5 -1 -n i 0 1 . Geometric Representation of a Complex Number- in class. Definition Addition and multiplication of complex numbers and also multiplication by reals are given by x y x y x x y y x y x y xx -yy xy x y x y x y . Geometric Representation of Addition- in class. Multiplication later Examples a 3 4 3 0 4 0 7 0 7 b 3x4 3 0 4 0 12-0 0 12 0 12 c 0 y y 0 1 yi which we also write as iy . d In general z x y x 0 0 y x iy. z x iy e Also i2 0 1 0 1 -1 0 -1. i2 -1 g 4 - 3i 4 -3 . Note In view of d above from now on we shall write the complex number x y as x iy. Definitions The complex conjugate z of the complex number z x iy given by z x - iy. The magnitude Izl of z x iy is given by Izl x2 y2 . Examples and Geometric Representation of Conjugation and Magnitude - in class. Notes 1. z z x iy x-iy 2x 2Re z . Therefore Re z 2 z z 2 Sirnpo PDjLMptseand SP 4nieg s tP ed V ersioYi1eh wwwsimpondflcom K. 2 .2 2 2 2 2 2. Note that zz X iy X-iy x-iy X y lzl 3. If z 0 then z has a multiplicative inverse. Why because 4 lzl2 1 H zlzl2 Izl2 Izl2 zz Izl2 -1 _ -2-z lzl2 Examples a 1 -i c I 1 Ể 1-i 4 1 i V2 V2 1 3-4i b 3 4i - 25 d cos0 isin0 cos -0 isin -0 4. There is also the Triangle Inequality lz1 z2l lz1l lz2l. Proof We square both sides and compare them. Write z1 X1 iyi and z2 x2 iy2. Then lzi z2l2 X1 X2 2 yi y2 2 X12 X22 .