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ECE 6382 Notes 1 Introduction to Complex Variables Fall 2017
David R. Jackson Notes 1 Introduction to Complex Variables Notes are adapted from D. R. Wilton, Dept. of ECE
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Some Applications of Complex Variables
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)
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Complex Arithmetic and Algebra
A complex number z may be thought of simply as an ordered pair of real numbers (x, y) with rules for addition, multiplication, etc. z x y r Argand diagram 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.
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Complex Arithmetic and Algebra (cont.)
y z1 z2 x1 y1 x2 y2 z1+ z2 -z2 z1- z2 Division is kind of messy in rectangular coordinates!
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Complex Arithmetic and Algebra (cont.)
Geometrical interpretation of addition and subtraction of complex numbers: Geometrically, this works the same way and adding and subtracting two-dimensional vectors. x y z1 z2 x1 y1 x2 y2 z1+ z2 -z2 z1- z2 Note: We can multiply and divide complex numbers. We cannot divide two-dimensional vectors. We can multiply two-dimensional vectors in different ways (dot product and cross product).
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Complex Arithmetic and Algebra (cont.)
z x y r z*
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Euler’s Formula
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Application to Trigonometric Identities
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Application to Trigonometric Identities (cont.)
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DeMoivre’s Theorem
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Roots of a Complex Number
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Roots of a Complex Number (cont.)
z x y u v w Re Im Cube root of unity
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