matematika

MECHANICAL AND INDUSTRIAL ENGINEERINGUNIVERSITAS GADJAH MADA

TKM 2104 Matematika Teknik 2 (4 SKS)Engineering Mathematics 2Semester 2, TA 2014/2015

Lecture 3-4Derivative of Complex Numbers

Dosen: Indro Pranoto, S.T., M.Eng.Email: [email protected]

Lecture 3-4 Derivative of Complex Numbers

Cauchy-Riemann Equations

Materi Kuliah:

13. Complex Numbers and Functions, Complex DifferentiationComplex Numbers and Their Geometric Representation (L1)Polar Form of Complex Numbers, Powers and Roots (L1-L2) Derivative Analytic Function (L3)CauchyRiemann Equations, Laplaces Equation (L3)Exponential Function (L4) Trigonometric and Hyperbolic Functions, Eulers Formula (L4)

Complex Analysis

Trigonometric and Hyperbolic Functions, Eulers Formula (L4) Logarithm, General Power and Principal Value (L4)

14. Complex Integration Line Integral in the Complex Plane (L5)Cauchys Integral Theorem (L5)Cauchys Integral Formula (L5)

15. Power Series and Taylor SeriesSequences, Series, Convergence Tests (L6)Power Series (L6) Functions Given by Power Series (L7) Taylor and Maclaurin Series (L7)Uniform Convergence (Optional) (L7*)

16. Laurent Series and Residue Integration Laurent Series (L8)Singularities and Zeros. Infinity (L8)Singularities and Zeros. Infinity (L8)Residue Integration Method (L9)Residue Integration of Real Integrals (L9)

17. Conformal MappingGeometry of Analytic Functions: Conformal Mapping (L10) Linear Fractional Transformations (Mbius Transformations) (L10)Special Linear Fractional Transformations (L11)Conformal Mapping by Other Functions (L11)Riemann Surfaces (Optional) (L11*)

Derivative of Complex NumbersCircles and Disks. Half-Planes

Complex Function

Recall from calculus that a real function f defined on a set S of real numbers (usually an interval) is a rule that assigns to every x in S a real number f(x), called the value of f at x.

In complex, S is a set of complex numbers. And a function f defined on S is a rule that assigns to every z in S a complex number w, called the value of f at z

z varies in S and is called a complex variable

The set S is called the domain of definition of f or, briefly, the domain of f. (In most cases S will be open and connected, thus a domain as defined just before.)

is a complex function defined for all z; that is, its domain S is the whole complex plane.

The set of all values of a function f is called the range of f.

Example:

This shows that a complex function f(z) is equivalent to a pair of realfunctions u(x,y) and v(x,y), each depending on the two real variables x and y.

Example:

Example:

Limit, Continuity

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Note that by definition of a limit this implies that f(z) is defined in someneighborhood of z0.

f(z) is said to be continuous in a domain if it is continuous at each point ofthis domain.

Derivative

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Example

The differentiation rules are the same as in real calculus, since their proofs are literally the same. Thus for any differentiable functions f and g and constant c we have

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Analytic Functions

CauchyRiemann Equations. Laplaces Equation.To do complex analysis on any complex function, we require that function tobe analytic on some domain that is differentiable in that domain.

f is analytic in a domain D if and only if the first partial derivatives of u and vsatisfy the two CauchyRiemann equations.

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Theorem 1

By assumption, the derivative f(z) at z exists. It is given by

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(5)Theorem 2

Example:

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