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Ç°ÑÔchapter i  complex numbers and functions  1  complex numbers    1.1  complex number field    1.2  complex plane    1.3  modulus, conjugation, argument, polar representation    1.4  powers and roots of complex numbers    exercises  2  regions in the complex plane    2.1  some basic concept    2.2  domain and jordan curve    exercises  3  functions of a complex variable    3.1  the concept of functions of a complex variable    3.2  limits and continuous    exercises  4  the extended complex plane and the point at infinity    4.1  the spherical representation, the extended complex plane    4.2  some concepts in the extended complex plane    exerciseschapter ii  analytic functions  1  the concept of the analytic function    1.1  the derivative of the functions of a complex variable    1.2  analytic functions    exercises  2  cauchy-riemann equations    exercises  3  elementary functions    3.1  the exponential function    3.2  trigonometric functions    3.3  hyperbolic functions    exercises  4  multi-valued functions    4.1  the logarithmic function    4.2  complex power functions    4. 3  inverse trigonometric and hyperbolic functions    exerciseschapter iii  complex integration  1  the concept of contour integrals    1.1  integral of a complex function over a real interval    1.2  contour integrals    exercises    cauchy-goursat theorem    2.1  cauchy theorem    2.2  cauchy integral formula    2.3  derivatives of analytic functions    2.4  liouville's theorem and the fundamental theorem of algebra    exercises    harmonic functions    exerciseschapter iv  series  1  basic properties of series    1.1  convergence of sequences    1.2  convergence of series    1.3  uniform convergence    exercises  2  power series    exercises  3  taylor series    exercises  4  laurent series    exercises  5  zeros of an analytic functions and uniquely determined analytic    functions    5.1  zeros of analytic functions    5.2  uniquely determined analytic functions    5.3  maximum modulus principle    exercises  6  the three types of isolated singular points at a finite point    exercises  7  the three types of isolated singular points at a infinite point    exerciseschapter v  calculus of residues  1  residues    1.1  residues    1.2  cauchy's residue theorem    1.3  the calculus of residue    exercises  2  applications of residue    2.1  the type of definite integral ¡õ    2.2  the type of improper integral ¡õ    2.3  the type of improper integral ¡õ    exercises  3  argument principle    exerciseschapter vi  conformal mappings  1  analytic transformation    1.1  preservation of domains of analytic transformation    1.2  conformality of analytic transformation    exercises  2  rational functions    2.1  polynomials    2.2  rational functions    exercises  3  fractional linear transformations    exercises  4  elementary conformal mappings    exercises  5  the riemann mapping theorem    exercisesappendix  appendix 1  appendix 2answersbibliography