The development of the theory of functions of a complex variable provides a fascinating example of how initial vague and uncertain ideas can be turned into a powerful and rigorous theory with far-ranging applications. The key aspects of this development are studied, from the first attempts to satisfactorily define and manipulate complex numbers, to the immense contributions of mathematicians such as Cauchy in the 19th century, and finally to the modern era, where the advent of high speed computers has brought mathematics to a wide audience.

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