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 used Visual Complex Analysis by T. Needham. Not only this book introduces reader to the intricacies of the Complex Analysis, but it gives a very intuitive picture and reasoning for visual representation of the subject. Well written, concise, and clear.
Complex Variable and Applications, by Ruel V. Churchill & James Ward Brown. It's known as "Churchill". It starts from the very beginning (what complex numebrs are) but it quickly steps into analytical functions, integrals and series. It covers everything quickly but enough. I'd say it's the main book for physicists about this subject.
My course at the TU Delft used Applied Complex Variables for Scientists and Engineers by Yue Kuen Kwok. I liked it because as such books go it is relatively short, quite readable, and full of examples and problems. What I especially liked is that it always stays very close to real applications but it refuses to sacrifice rigor to do this.
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.
As if aware that our minds are blown, section 5 returns back to the more tame idea of Taylor series and extends it to Laurent series and their convergence, and analytically continuing a power series outside of its radius of convergence to cover its entire Riemann surface, and section 6 returns to the Cauchy integral theorem with the residue calculus and the classification of poles, so there is some time for the dense mathematics to sink in. One recurring theme is trying to describe 2D fluid flow. There is also some discussion in these chapters about Fourier transforms and the like, with practice on adding contours that integrate to zero so that we can use the contour-deformation tricks that Cauchy gives us to get sums of residues. Section 7 was somewhat forgettable but introduced the Laplace transform which was nice, but section 8 on conformal mappings actually became very useful for me later on, as it turns out that the mathematics of bilinear transformations discussed in section 8.2 can be reappropriated to describe proper Lorentz transformations -- in 2-spinor calculus you find out that every Lorentz transform acts on the incoming and outgoing light rays from an observer as a bilinear tranform of a point on the complex plane, this point turns out to be a stereographic projection (see section 1!) of its location on the sphere of the sky. You get a really slick derivation both that accelerating observers must see the distant stars and galaxies all "crowd into" one point in the sky, and that their emitted light must "crowd into" the direction that they're going (so-called "relativistic beaming"). That was beyond the scope of the course, but the point is that I've seen this mathematics recur in a couple places now. (The latest was a use in continued-fraction calculators.)