Reference textbook –D.G. Zill and P.D. Shanahan, A First ......Complex Plane Reference textbook...
Transcript of Reference textbook –D.G. Zill and P.D. Shanahan, A First ......Complex Plane Reference textbook...
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1301610313016103Mathematics Mathematics 33#11 Analytic Functions and Integration in the Complex Plane
Reference textbook – D.G. Zill and P.D. Shanahan, A First Course in Complex Analysis with Applications, 2nd ed., The Jones and Bartlett Publisher, 2009
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Content of this lectureContent of this lecture
� Differentiability
� Analyticity
� Cauchy-Riemann equations
� Integration in the complex plane� Integration in the complex plane
Reference textbook – D.G. Zill and P.D. Shanahan, A First Course in Complex Analysis with Applications, 2nd ed., The Jones and Bartlett Publisher, 2009
Complex Analysis 2
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DIFFERENTIABILITYDIFFERENTIABILITY
Complex Analysis 3
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DifferentiabilityDifferentiability
� Suppose z = x + iy and z0 = x0 + iy0;
� then the change in z0 is the difference
∆z = z – z0 or
∆z = (x – x ) + i(y – y ) = ∆x + i ∆y∆z = (x – x0) + i(y – y0) = ∆x + i ∆y
� If a complex number w = f(z) is defined at zand z0, then the corresponding change in function is the difference
∆w = f(z0 + ∆z) – f(z0).
Complex Analysis 4
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DifferentiabilityDifferentiability
� The derivative of the function f is defined in terms of a limit of the difference quotient ∆w/∆z as ∆z � 0.
Complex Analysis 5
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DifferentiabilityDifferentiability
� If the limit in (1) exists, then the function fis said to be differentiable at z0.
� Two other symbols denoting the derivative � Two other symbols denoting the derivative of w = f(z) are w' and dw/dz.
� If the latter notation is used, then the value of a derivative at a specified point z0 is written .
Complex Analysis 6
0zzdz
dw
=
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Example Example 11
� Use the definition 3.1 to find the derivative of f(z) = z2 – 5z.
Complex Analysis 7
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Rules of differentiationRules of differentiation
Complex Analysis 8
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Rules of differentiationRules of differentiation
� The power rule for differentiation of powers of z is also valid:
� Combining (6) and (7) gives the power rule for functions:
Complex Analysis 9
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Example Example 22
� Differentiate:
a) f(z) = 3z4 – 5z3 +2z
b) f(z) = z2/(4z + 1)b) f(z) = z2/(4z + 1)
c) f(z) = (iz2 + 3z)5
Complex Analysis 10
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A function that is nowhere A function that is nowhere differentiabledifferentiable� For a complex function f to be differentiable at a point z0, we know from the preceding lecture that the limit
zfzzf −∆+ )()(lim 00
must exist and equal the same complex number from any direction.
� The limit must exist regardless how ∆zapproaches 0.
Complex Analysis 11
z
zfzzfz ∆
−∆+→∆
)()(lim 00
0
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A function that is nowhere A function that is nowhere differentiabledifferentiable� This means that in complex analysis, the requirement of differentiability of a function f(z) at a point z0 is a far greater demand than in real calculus of function demand than in real calculus of function f(x) where we can approach a real number x0 on the number line from only two directions.
� An example of a complex function that is not differentiable is f(z) = x + 4iy.
Complex Analysis 12
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Example Example 33
� Show that the function f(z) = x + 4iy is not differentiable at any point z.
Complex Analysis 13
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ANALYTICITYANALYTICITY
Complex Analysis 14
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Analytic functionsAnalytic functions
� Even though the requirement of differentiability is a stringent demand, there is a class of functions that is of great importance whose members satisfy even importance whose members satisfy even more severe requirements.
� These functions are called analytic functions.
Complex Analysis 15
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Analytic functionsAnalytic functions
� A function f is analytic in a domain D if it is � A function f is analytic in a domain D if it is analytic at every point in D.
� The phrase “analytic on a domain D” is also used but we shall call a function f that is analytic throughout a domain D holomorphicor regular.
Complex Analysis 16
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Analytic functionsAnalytic functions
� Note: analyticity at a point is not the same � Note: analyticity at a point is not the same as differentiability at a point!
� Analyticity at a point is a neighborhood property; in other words, analyticity is a property that is defined over an open set.
Complex Analysis 17
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Analytic functionsAnalytic functions
� As an example, the function f(z) = |z|2 is � As an example, the function f(z) = |z| is differentiable at z = 0 but is not differentiable anywhere else.
� Even though f(z) = |z|2 is differentiable at z = 0, it is not analytic at that point because there exists no neighborhood of z = 0 throughout which f is differentiable; hence, f(z) = |z|2 is nowhere analytic!
Complex Analysis 18
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ExampleExample
� Show that the function f(z) = |z|2 is differentiable only at z = 0.
Complex Analysis 19
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Entire functionsEntire functions
� A function that is analytic at every point z in the complex plane is said to be an entire function.
� According to the differentiation rules (2), � According to the differentiation rules (2), (3), (5), and (7), we can conclude that polynomial functions are differentiable at every point z in the complex plane and rational functions are analytic throughout any domain D that contain no points at which the denominator is zero.
Complex Analysis 20
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Entire functionsEntire functions
Complex Analysis 21
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Differentiability & continuityDifferentiability & continuity
� As in real analysis, if a function f is differentiableat a point, the function is necessarily continuous at at a point, the function is necessarily continuous at the point.
� Of course the converse is not true; continuity of a function f at a point does not guarantee that f is differentiable at the point.
� Example, f(z) = x + 4iy is continuous everywhere but is nowhere differentiable.
Complex Analysis 22
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L’Hopital’s ruleL’Hopital’s rule
� L’Hopitals’rule for computing limits of the indeterminate form 0/0, carries over to complex analysis.
Complex Analysis 23
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ExampleExample
� Using L’Hopital’s rule to compute
izz
zziz 10
54lim
3
2
2 −−+−
+→
Complex Analysis 24
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CAUCHYCAUCHY--RIEMANN RIEMANN EQUATIONSEQUATIONSEQUATIONSEQUATIONS
Complex Analysis 25
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CauchyCauchy--Riemann equationsRiemann equations
� Here, we sahll learn a test for analyticity of a complex function f(z) = u(x, y) + iv(x, y) that is based on partial derivatives of its real and imaginary parts u and v.its real and imaginary parts u and v.
Complex Analysis 26
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CauchyCauchy--Riemann equationsRiemann equations
Complex Analysis 27
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CauchyCauchy--Riemann equationsRiemann equations
Complex Analysis 28
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CauchyCauchy--Riemann equationsRiemann equations
Complex Analysis 29
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ExampleExample
� Using Cauchy-Riemann equation to show that f(z) = x + 4iy is not analytic.
Complex Analysis 30
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ExampleExample
� Verify that the polynomial function f(z) = z2 + z satisfies Cauchy-Riemann equations for all z.
Complex Analysis 31
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ExampleExample
� Show that the complex function f(z) = 2x2
+ y + i(y2 – x) is not analytic at any point.
Complex Analysis 32
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Criterion for analyticityCriterion for analyticity� By themselves, the Cauchy-Riemann equations do not ensure analyticity of a function f(z) = u(x, y) + iv(x, y) at a point z = x + iy.
It is possible for the Cauchy-Riemann � It is possible for the Cauchy-Riemann equations to be satified at z and yet f(z)may not be differentiable at z, or
� f(z) may be differentiable at z but nowhere else.
� In either case, f is not analytic at z.Complex Analysis 33
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Criterion for analyticityCriterion for analyticity
� However, when we add the condition of continuity to u and v and to the four partial derivatives , , and , it can be shown that the
xu ∂∂ yu ∂∂ xv ∂∂yv ∂∂and , it can be shown that the
Cauchy-Riemann equations are not only necessary but also sufficient to guarantee analyticity of f(z) at z.
Complex Analysis 34
yv ∂∂
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Criterion for analyticityCriterion for analyticity
Complex Analysis 35
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ExampleExample
� Use Theorem 3.5 to check the analyticity of
2222)(
yx
yi
yx
xzf
+−
+=
Complex Analysis 36
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CauchyCauchy--Riemann equations in polar Riemann equations in polar coordinatecoordinate� Indeed, the form f(z) = u(r, ɵ) + iv(r, ɵ) is often more convenient to use.
� In polar coordinate, the Cauchy-Riemann equations becomesequations becomes
and
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INTEGRATION IN THE INTEGRATION IN THE COMPLEX PLANECOMPLEX PLANECOMPLEX PLANECOMPLEX PLANE
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Curve revisitedCurve revisited
� Suppose the continuous real-valued functions x = x(t), y = y(t), a ≤ t ≤ b, are parametric equations of a curve Cin the complex plane.
If we use these equations as the real � If we use these equations as the real and imaginary parts in z = x + iy, we can describe the point z on C by means of a complex-valued function of a real variable t called parametrization of C:
Complex Analysis 39
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Complex integralComplex integral
� An integral of a function f of a complex variable z that is defined on a contour (piecewise smooth curve) C is denoted by (piecewise smooth curve) C is denoted by
and is called a complex integral or contour integral.
Complex Analysis 40
∫C dzzf )(
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Complex integralComplex integral
� To evaluate a contour integral , let us write (2) in an abbreviated form.
∫C dzzf )(
� The intepretation is
Complex Analysis 41
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Complex integralComplex integral
� Specifically, the right-hand side becomes
Complex Analysis 42
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Complex integralComplex integral� If we use the complex-valued function (1) to describe the contour C, then (10) is the same as when the integrand is∫ ′
b
adttztzf )())((
� Thus we arrive at a practical means of evaluating a contour integral.
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ExampleExample
� Evaluate , where C is given by x = 3t, y = t2, -1 ≤ t ≤ 4.
∫C dzz
Complex Analysis 44
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ExampleExample
� Evaluating , where C is the circle x = cos t, y = sin t, 0 ≤ t ≤2π.
∫C dzz
Complex Analysis 45
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Properties of contour integralProperties of contour integral
Complex Analysis 46
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Bounding theoremBounding theorem
� Sometimes, it may be useful to find an upper bound for the modulus or absolute value of a contour integral.
� The following theorem is proved by using � The following theorem is proved by using the form of the triangle inequality.
Complex Analysis 47
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ExampleExample
� Find an upper bound for the absolute value of where C is the circle |z| = 4.
∫ +C
z
dzz
e
1
Complex Analysis 48