complexVariables

complexVariables - Complex Variables 020701 F. Porter...

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Complex Variables 020701 F. Porter Revision 091006 1 Introduction This note is intended as a review and reference for the basic theory of complex variables. For further material, and more rigor, Whittaker and Watson is recommended, though there are very many sources available, including a brief review appendix in Matthews and Walker. 2 Complex Numbers Let z be a complex number , which may be written in the forms: z = x + iy (1) = re , (2) where x , y , r ,and θ are real numbers. The quantities x and y are referred to as the real and imaginary parts of z , respectively: x = < ( z ) , (3) y = = ( z ) . (4) The quantity r is referred to as the modulus or absolute value of z , r = | z | = q x 2 + y 2 , (5) and θ is called the argument, θ =arg( z ), or the phase , or simply the angle of z . We have the transformation between these two representations: x = r cos θ, (6) y = r sin (7) and ±nally also θ =tan 1 ( y/x ) , (8) with due attention to quadrant. Noticing that e = e i ( θ +2 ) ,where n is any integer, we say that the principal value of arg z is in the range: π< arg z π. (9) 1
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y x r z z * θ θ Figure 1: Complex number and its complex conjugate. The complex conjugate , z ,o f z is obtained from z by changing the sign of the imaginary part: z = x iy = re . (10) The product of two complex numbers, z 1 and z 2 ,isg ivenby : z 1 z 2 = r 1 e 1 r 2 e 2 = r 1 r 2 e i ( θ 1 + θ 2 ) =( x 1 + iy 1 )( x 2 + iy 2 )=( x 1 x 2 y 1 y 2 )+ i ( x 1 y 2 + x 2 y 1 ) . (11) Notice that zz = x 2 + y 2 = | z | 2 . (12) It is also interesting to notice that in the product: z 1 z 2 x 1 x 2 + y 1 y 2 ) i ( x 1 y 2 x 2 y 1 ) , (13) the real part looks something like a “scalar product” of two vectors, and the imaginary part resembles a “cross product”. 3 Complex Functions of a Complex Variable We are interested in (complex-valued) functions of a complex variable z .In particular, we are especially interested in functions which are single-valued, continuous, and possess a derivative in some region. 2
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Defning a suitable derivative requires some care. Start with the defnition For real Functions oF a real number: f 0 ( x )= df dx ( x ) = lim Δ x 0 f ( x x ) f ( x ) Δ x . (14) But in the complex case we have real and imaginary parts to worry about. ±irst, defne what we mean by a limit. Let f ( z ) be a single-valued Function defned at all points in a neighborhood oF z 0 (except possibly at z 0 ). Then we say that f ( z ) w 0 as z z 0 , or lim z z 0 f ( z w 0 , iF, For every ±> 0, there exists a δ> 0 such that (±ig. 2): | f ( z ) w 0 | z satisFying 0 < | z z 0 | . (15) Note that we have not required “ f ( z 0 )” to be defned, in order to defne the limit (±ig. 3). y x z 0 d ±igure 2: Circle oF radius δ about z 0 . "f(z)" "z" z 0 ±igure 3: ±unction not defned at z 0 . However, in order to defne the derivative at z 0 ,werequ i re f ( z 0 )tobe defned. IF lim z z 0 = f ( z 0 ), where the limit exists, then we say that f ( z )is continuous at z 0 . In general f ( z ) is complex, and we may write: f ( z u ( x, y )+ iv ( x, y ) , (16) 3
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where u and v are real. Then lim z z 0 = f ( z 0 ) implies lim x x 0 ,y y 0 u ( x, y )= u ( x 0 0 ) , (17) lim x x 0 y 0 v ( x, y v ( x 0 0 ) , (18) where the path of approach to the limit point must lie within the region of deFnition. We may thus deFne continuity to the boundary of a closed region,
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complexVariables - Complex Variables 020701 F. Porter...

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