# Complex Variable Book Suggestion

What book should I choose to learn complex analysis as a physics Undergrad. I only want to use one book which will contain everything I need.

I also liked the pacing. Coming into section 1 you are introduced from-scratch to complex numbers, with defined terms like "modulus" and "argument". You are shown how they densely summarize cosine and sine rules, how they can be used for electrical circuits (impedances, basically) and their connection to stereographic projections. In section 2 you start to learn how to differentiate complex numbers and you are introduced to the Cauchy-Riemann conditions and the notion of an analytic function -- but also how they can be used to solve the heat equation and other places where harmonic functions are useful. Then section 3 gently guides the reader to the correct complex understanding of the exponential function, the logarithm, the trigonometric functions, and the hyperbolic trigonometric functions, as well as to fractional (and real) powers. The Riemann surfaces that a multivalued function is "really" defined over are discussed, as are the "branch cuts" that slice-and-dice them down into the more familiar "branch sheets" that work as little well-behaved complex planes. Only after all of this gradual buildup to we get to the really "intense" stuff in section 4 about complex integration, the Cauchy integral theorem and contour deformation, the Cauchy integral formula and its implication that complex-differentiable-once (in a neighborhood) implies infinitely-often-complex-differentiable (in that neighborhood), and the theorems of Liouville (analytic over all of $\mathbb C$= either unbounded or constant) and of maximums (analytic on a bounded domain = maximum is on the boundary). There are some applications to vector fields as well in there.