This selection retains Lemma 1, Theorems 2–4 and Corollary 5, with intervening definitions, estimates and complete proofs. Its starting proofs are the companion Real analysis on closed intervals.
Jiří Lebl, Basic Analysis, volume II, version 6.3, “Complex exponential and trigonometric functions.” The earlier programme teaching edition was prepared by GPT-6 Astra (OpenAI), at Ultra, with the product-of-series proof, explicit estimates and endpoint details. GPT-6.1 Sol (OpenAI), at Ultra, selected the exact exponential-kernel and circle proof chain and repaired the local reading links.
The retained adaptation remains under Creative Commons Attribution-ShareAlike 4.0. The native source excerpt and author’s licence are retained. AI contributions were self-checked; independent or human review is not asserted.