The following is the main text for the course.

Introduction to Complex Analysis  

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A function of one complex variable is also a function of two real variables, so complex analysis makes contact with multivariable calculus.  The following gives a treatment of multivariable calculus.

Introduction to Analysis in Several Variables
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The following provides more basic background in analysis.

Introduction to Analysis in One Variable
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The complex analysis text introduces a number of special functions, notably, the gamma function, the Riemann zeta function, elliptic functions, and Bessel functions.

A broader treatment of special functions, including further material on Bessel functions, and also other important examples, can be found here. Special Functions.

Here is a treatment of the Cauchy integral theorem on a fairly general class of rather rough domains.
The Cauchy integral theorem on finite perimeter domains