This book is a translation of the forthcoming fourth edition of our German A book “Funktionentheorie I” (Springer 2005). The translation and the LT X E ?les have been produced by Dan Fulea. He also made a lot of suggestions for improvement which in?uenced the English version of the book. It is a pleasure for us to express to him our thanks. We also want to thank our colleagues Diarmuid Crowley, Winfried Kohnen and J ̈ org Sixt for useful suggestions c- cerning the translation. Over the years, a great number of students, friends, and colleagues have c- tributed many suggestions and have helped to detect errors and to clear the text. The many new applications and exercises were completed in the last decade to also allow a partial parallel approach using computer algebra systems and graphictools,whichmayhaveafruitful,powerfulimpactespeciallyincomplex analysis. Last but not least, we are indebted to Clemens Heine (Springer, Heidelberg), who revived our translation project initially started by Springer, New York, and brought it to its ?nal stage.