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A First Course in Complex Analysis with Applications (Jones and Bartlett Publishers Series in Mathematics) 2nd Edition
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That being said, I've struggled greatly to find a text that was self contained. I own a large number, both pitched to math majors and engineers. I have Needham, Silverman, Ablowitz and Fokas, Saff and Snyder, Palka, D'Angelo, Bak and Newman, and now Zill and Shanahan.
Not all of the these texts are terrible, but Zill and Shanahan is the most self-contained and learning from this is most expedient. It is the only book I've seen that is structured like one of the now commonplace textbooks in the calculus sequence e.g. the Stewart series. For a subject like Complex analysis I think this structure is very appropriate as an introduction. The next closest text in coherency is the Ablowitz and Fokas. Both have greatly enhanced my understanding. Sometimes I supplement the former with problems from the latter, which tend to be a bit more challenging.
For a student with little experience with complex numbers/analysis, I don't think there is a better substitute for a first course. This is the safest bet.
Incidentally, this is perfectly suitable for math majors. It's rigorous enough and the problems are pitched in a range of difficulty. Shouldn't be deprecated because it has "applications" in the title.
It has only one disadvantage, the price. However, it is a good investment. I do recommend this one.
If you doubt, look inside!