Introduction to Complex Analysis����Dr. Deepali
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1.Complex Number
We here go through the complex algebra briefly.
A complex number z = (x,y) = x + iy, Where
We will see that the ordering of two real numbers (x,y) is significant,
i.e. in general x + iy ≠ y + ix
X: the real part, labeled by Re(z); y: the imaginary part, labeled by Im(z)
Three frequently used representations:
(1) Cartesian representation: x+iy
(2) polar representation, we may write
z=r(cos θ + i sinθ) or
r – the modulus or magnitude of z
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r – the modulus or magnitude of z
The relation between Cartesian
and polar representation:
The choice of polar representation or Cartesian representation is a
matter of convenience. Addition and subtraction of complex variables
are easier in the Cartesian representation. Multiplication, division, powers, roots are easier to handle in polar form,
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From z, complex functions f(z) may be constructed. They can be written
f(z) = u(x,y) + iv(x,y)
in which v and u are real functions.
For example if , we have
The relationship between z and f(z) is best pictured as a mapping operation, we address it in detail later.
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Using the polar form,
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Function: Mapping operation
x
y
Z-plane
u
v
The function w(x,y)=u(x,y)+iv(x,y) maps points in the xy plane into points
in the uv plane.
We get a not so obvious formula
Since
Complex Conjugation: replacing i by –i, which is denoted by (*),
We then have
Hence
Note:
ln z is a multi-valued function. To avoid ambiguity, we usually set n=0
and limit the phase to an interval of length of 2π. The value of lnz with
n=0 is called the principal value of lnz.
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Special features: single-valued function of a
real variable ---- multi-valued function
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Another possibility
Question:
1.2 Cauchy – Riemann conditions
Having established complex functions, we now proceed to
differentiate them. The derivative of f(z), like that of a real function, is
defined by
provided that the limit is independent of the particular approach to the
point z. For real variable, we require that
Now, with z (or zo) some point in a plane, our requirement that the
limit be independent of the direction of approach is very restrictive.
Consider
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,
Let us take limit by the two different approaches as in the figure. First,
with δy = 0, we let δx🡪0,
Assuming the partial derivatives exist. For a second approach, we set
δx = 0 and then let δy🡪 0. This leads to
If we have a derivative, the above two results must be identical. So,
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,
These are the famous Cauchy-Riemann conditions. These Cauchy-
Riemann conditions are necessary for the existence of a derivative, that
is, if exists, the C-R conditions must hold.
Conversely, if the C-R conditions are satisfied and the partial
derivatives of u(x,y) and v(x,y) are continuous, exists. (see the proof
in the text book).
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Analytic functions
If f(z) is differentiable at and in some small region around ,
we say that f(z) is analytic at
Differentiable: Cauthy-Riemann conditions are satisfied
the partial derivatives of u and v are continuous
Analytic function:
Property 1:
Property 2: established a relation between u and v
Example: