# University of Toronto at Scarborough Department of Computer...

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University of Toronto at Scarborough Department of Computer and Mathematical Sciences MAT C34F 2013/14 Complex Variables Instructor: Prof. L. Jeffrey Office: IC-474 Telephone: (416)287-7265 Email: [email protected] Review of Material for Midterm on November 1, 2013 Assigment 1 1. A number z = x + iy in the complex plane can be written in polar coordinates as z = re where r 0 is a real number r = x 2 + y 2 and e = cos( θ ) + i sin( θ ) where cos( θ ) = x/r and sin( θ ) = y/r . The equation z n = 1 has n roots: these are z = e 2 πim/n where m = 0 , . . . , n 1. 2. The complex conjugate of z is ¯ z = x iy. 3. The modulus of z is | z | = x 2 + y 2 . 4. The logarithm of a complex number z = re is [Log( z )] = { log( r ) + : θ [Arg( z )] } where the argument of z is [Arg( re )] = { θ + 2 πn } for n = 0 , ± 1 , ± 2 , . . . 1

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1. A complex-valued function f is differentiable at z if df dz ( z )= lim h 0 f ( z + h ) f ( z ) h exists. Here, the limit is taken as the complex number h tends to zero. If one considers h tending to zero along a fixed direction in the complex plane (in other words h = re where r 0 but θ remains constant) the limit must give the same value regardless of the value of the angle θ . 2. A complex valued function f is holomorphic at z 0 if f is differentiable at all z in an open set containing z 0 . 3. If a function f ( z ) = u ( x, y ) + iv ( x, y ) is differentiable at z , then it satisfies the Cauchy-Riemann equations ∂u ∂x = ∂v ∂y ; ∂v ∂x = ∂u ∂y .
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