## Functions of a Complex Variable: With Applications |

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Page 8

The equation z = x(t)+iy(t), where w(t) and y(t) are real continuous functions of the

Argand diagram which is called a continuous arc. A point z1 is a multiple point of

the ...

The equation z = x(t)+iy(t), where w(t) and y(t) are real continuous functions of the

**real variable**t, defined in the range a stSB, determines a set of points in theArgand diagram which is called a continuous arc. A point z1 is a multiple point of

the ...

Page 10

the complex variable z is exactly the same thing as a complex function w(x, y)+iv(

x, y) of two

of continuity is exactly the same as that for functions of a

the complex variable z is exactly the same thing as a complex function w(x, y)+iv(

x, y) of two

**real variables**a and y. For functions defined in this way, the definitionof continuity is exactly the same as that for functions of a

**real variable**.Page 22

For real values of y we have is – Go (iy)" oo k gok o yokol “-of -2, (-) is +:(-1)*::::Hsi

() = cos y +i sin y : since the cosine and sine of the

the two power series on the right of (2). Hence e” = e^*** = e^e” = e^ cis y.

For real values of y we have is – Go (iy)" oo k gok o yokol “-of -2, (-) is +:(-1)*::::Hsi

() = cos y +i sin y : since the cosine and sine of the

**real variable**y are defined bythe two power series on the right of (2). Hence e” = e^*** = e^e” = e^ cis y.

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### Contents

PREFACE p vii | 1 |

CONFORMAL REPRESENTATION p | 32 |

SOME SPECIAL TRANSFORMATIONS p | 57 |

3 other sections not shown

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### Common terms and phrases

2tri analytic angle Argand diagram bilinear transformation boundary bounded Cauchy's theorem circle of centre closed contour complex numbers conformal transformation consider const constant continuous contour integral cosec cosh D.Sc defined definition denote derivatives differentiable domain Eacample ellipse essential singularity finite number follows given Hence infinite inside integral round interior point inverse isogonal Laplace's equation lemma limit point magnification many-valued function maps Möbius modulus neighbourhood obtain one-one parabola plane point at infinity pole of order polygon positive real axis power series prove radius rational function real and imaginary real numbers real variable rectangle region regular function Riemann surface roots ſ f(z)dz satisfied semicircle Show Similarly simple poles singularity of f(z sinh small circle ſº square straight line strip Taylor's theorem tends to zero unit circle upper half upper half-plane w-plane write z-plane corresponding z—zo z+iy