In the text "Function Theory of a Complex Variable", i'm having trouble proving the following relation in $(1.)$
$(1.)$
Prove that if $f$ is holomorphic on $\text{U} \subset \mathbb{C}$, then
$$\nabla(|f|^{2})=4 |\partial f / \partial z|^2)$$
$$\text{Lemma}$$
The following observations can b...
In transcendental number theory, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states that if α1, ..., αn are algebraic numbers which are linearly independent over the rational numbers ℚ, then eα1, ..., eαn are algebraically independent over ℚ; in other words the extension field ℚ(eα1, ..., eαn) has transcendence degree n over ℚ.
An equivalent formulation (Baker 1975, Chapter 1, Theorem 1.4), is the following: If α1, ..., αn are distinct algebraic numbers, then the exponentials eα1, ..., eαn are linearly independent over t...
