Chapter 20 Complex variables

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Chapter 20 Complex variables
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Chapter 20 Complex variables
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Chapter 20 Complex variables
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Chapter 20 Complex variables
20.2 Cauchy-Riemann relation
A function f(z)u(x,y)iv(x,y) is differentiable
and analytic, there must be particular connection
between u(x,y) and v(x,y)
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Chapter 20 Complex variables
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Chapter 20 Complex variables
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Chapter 20 Complex variables
20.3 Power series in a complex variable
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Chapter 20 Complex variables
20.4 Some elementary functions
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Chapter 20 Complex variables
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Chapter 20 Complex variables
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Chapter 20 Complex variables
20.5 Multivalued functions and branch cuts
A logarithmic function, a complex power and a
complex root are all multivalued. Is the
properties of analytic function still applied?
(A)
(A)
(B)
(B)
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Chapter 20 Complex variables
Branch point z remains unchanged while z
traverse a closed contour C about some point. But
a function f(z) changes after one complete
circuit.
Branch cut It is a line (or curve) in the
complex plane that we must cross , so the
function remains single-valued.
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Chapter 20 Complex variables
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Chapter 20 Complex variables
(A)
(B)
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Chapter 20 Complex variables
20.6 Singularities and zeros of complex function
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Chapter 20 Complex variables
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Chapter 20 Complex variables
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Chapter 20 Complex variables
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Chapter 20 Complex variables
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Chapter 20 Complex variables
20.10 Complex integral
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Chapter 20 Complex variables
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Chapter 20 Complex variables
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Chapter 20 Complex variables
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Chapter 20 Complex variables
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Chapter 20 Complex variables
20.11 Cauchy theorem
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Chapter 20 Complex variables
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Chapter 20 Complex variables
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Chapter 20 Complex variables
20.12 Cauchy