This is a revised edition of the chapter on Complex Variables, which was published few years ago in Part II of My Personal Study Notes in Advanced Mathematics. In this edition, I reproduced, refined, and enhanced all the calculations and graphics in a modern style of representation. In addition, I re-typed the cursive scripts of the personal notes and edited the typographic errors. In the editing process, I added plenty of comments on the underlying meanings of the arcane equations, such that the reader could discern the practical weight of each mathematical formula. In a way, I attempted to convey a personal sense and feeling on the significance and philosophy of devising a mathematical equation that transcends into real-life emulation. When equations deviate clearly from traditional algebraic patterns, I attempted to help the reader to understand how non-algebraic problems could be solved graphically and through programmed numerical iteration. The reader will find this edition dense with graphic illustrations, made even for the simplest configurations that should spare the reader the trouble of searching other references in order to infer any missing steps. In my view, detailed graphic illustrations could sooth the harshness of arcane mathematical jargon, as well as expose the merits of the assumptions contemplated in the formulation. In lieu of offering a dense textbook on Complex Variables, I opted to stick to my personal notes that give the memorable zest of a subject that could easily fade away in the mind when not frequently used 1. Introduction to complex numbers 2. Theory of functions of complex variables 2.1. The real and imaginary parts of a complex number 2.2. Complex number as vectors 2.3. Modulus and Argument of a complex number 2.4. Cartesian and Polar representation of complex numbers 2.5. Pure Real and Imaginary numbers 2.6. Representation of complex numbers on a Riemann Sphere 3. Algebraic operations on complex numbers 3.1. Addition of complex numbers 3.2. Reciprocal of a complex number 3.3. The product of complex numbers 3.4. Division of complex numbers 3.5. The nth roots of complex numbers 3.6. Conjugate complex numbers 3.7. Polynomials of complex numbers 3.8. Rational Functions of complex numbers 3.9. Exponential Functions of complex numbers 3.10. Hyperbolic Functions of complex numbers 3.11. Trigonometric Functions of complex numbers 3.12. Logarithmic Functions of complex numbers 3.13. Inverse Hyperbolic Functions of complex numbers 3.14. Inverse Trigonometric Functions of complex numbers 3.15. Complex Exponent Functions of complex numbers 3.16. Algebraic and Transcendental Functions of complex numbers 3.17. Definition of Limit of complex functions 3.18. Continuity of complex functions 4. Conformal Mapping 4.1. Jacobian or rates of change of x, y, on u, v 4.2. Example of mapping complex functions z ( x, y ) on a quadratic function w (u, v) 4.3. The Transformation of w (u, v) = exp z (x, y)] 4.4. The Transformation of w (u, v) = cosh z (x, y)] 5. Inverse Transformation 5.1. Inverse transformation of a line on a point on same line 5.2. Inverse transformation of a line on a point not on same line 5.3. Inverse transformation of a circle on a point on same circle 5.4. Inverse transformation of a circle on a point not on same circle 5.5. Inverse transformation of angle between two curves 5.6. Operations involved in the inverse transformation of curves 5.7. Summary of bilinear transformation of lines and circles 6. Series of complex numbers 6.1. Convergence of a series of complex numbers 6.2. Power Series of complex variables 6.3. Circle of convergence 6.4. Line Integral of a function of complex variables 7. Cauchy's Theorem 7.1. Isolating singularities with Cauchy's problems 7.2. Poles of singular functions 7.3. Cauchy's Theorem of Residue 7.4. Cauchy's Integral formula 8. Taylor's and LAURENT'S expansions