241-213 computer engineering mathematics

241-213 Computer Engineering Mathematics

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241-213241-213

Computer Engineering Computer Engineering MathematicsMathematics

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Chapter 21Chapter 21Complex Functions


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OUTLINE• Limits

• Continuity

• Derivatives


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Complex Functions

• A complex function is a function that is defined for complex numbers in some set S and takes on complex values.

• If C denotes the set of complex number, and

• f is such a function,

• then we write

• This means, f(z) is a complex number for each z in S.

• The set S is called the domain of f.

CSf :


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Example

• Let S consists of all z with |z| < 1• and define f(z) = z2 in S• then and f is a complex function.

• Often we define a function by some explicit expression in z, for example,

• In the absence of specifying S, we agree to allow all z which the expression for f(z) is defined

• This function defines for all complex z except 2i and -2i

CSf :

4)(

2

z

izzf


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Limits


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Limits


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Example 21.1

Let


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Limit Theorems• Many limit theorems from real calculus hold for complex

functions as well

• Suppose KzgLzf zzzz )(lim and )(lim00


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Limits of Real vs Complex functions

• Limits of real functions approach the point from 2 sides

• Limits of complex functions approach the point from any directions


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Continuity

• Any polynomial is continuous for all z

• Any rational function (quotient of polynomians) is continuous whenever its denominator is nonzero.

• If f is continuous at z0, so is |f|


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Continuity

• If {zn} is a sequence of complex numbers and each f(zn) is defined

• then {f(zn)} is also a complex sequence

• For example, if f(z) = 2z2 and zn = 1/n then f(zn) = 2/n2

• We claim that {f(zn)} converges if {zn} does, when f is continuous

• “Continuity preserves convergence of sequences”


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THEOREM 21.1

• Let be continuous, and let {zn} be a sequence of complex numbers in S. If {zn} converges to a number w in S, then {f(zn)} converges to f(w)

• A converse of Theorem 21.1

If f(zn) -> f(w) for every sequence {zn} of points of S converging to w, then f is continuous at w.

CSf :


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Bounded Function

• If f is bounded if there is a disk about the origin containing all the numbers f(z) for z in S

• A continuous function need not be bounded (look at f(z) = 1/z for z != 0)


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THEOREM 21.2

• Let . Suppose S is compact and f is continuous on S. Then f is bounded.

CSf :


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THEOREM 21.3

• Let be continuous and suppose S is compact. Then there are numbers zi and z2 in S such that, for all z in S

CSf :

)()()( 21 zfzfzf


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The Derivative of a Complex Function

• The reason for having S open in this definition is to be sure that there is some open disk about z0 throughout which f(z) is defined


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Example 21.2


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Derivative Formulas

• If n is a positive integer

• and f(z) = zn

• then f’(z) = nzn-1

• if f(z) = sin(z) then f’(z) = cos(z)


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Derivative Formulas


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Chain Rule


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Example


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THEOREM 21.4


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Cauchy-Riemann Equations

• We will derive a set of partial differential equations that must be satisfied by the real and imaginary parts of a differentiable complex function.


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Example 21.4


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Example 21.5


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THEOREM 21.5

Cauchy-Riemann Equations


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Proof


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Example


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Example


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Note


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Example 21.8


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Example 21.8 (cont.)


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THEOREM 21.6


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THEOREM 21.7


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241-213 computer engineering mathematics

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