Intereting Posts

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$x^5 + y^2 = z^3$

Plain and simple: Can each entire function $f$ without zeros be written in the form $e^g$ for some entire function $g$? What would be a counter-example?

- Physical or geometric meaning of complex derivative
- Solve $\cos(z)=\frac{3}{4}+\frac{i}{4}$
- Integrability of $f(t) =\frac{2^{\frac{it+1}{1.5}}}{2^{\frac{it+1}{2}}} \frac{\Gamma \left( \frac{it+1}{1.5} \right) }{\Gamma \frac{ it+1}{2} }$
- Cauchy's argument principle, trouble working simple contour integral
- A criterion for the existence of a holomorphic logarithm of a holomorphic function
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- Find the root of the polynomial?
- showing certain vertices form an equilateral triangle
- Composition of Analytic Functions
- The Duplication Formula for the Gamma Function by logarithmic derivatives.

Every zero-free holomorphic function $f$ on a simply connected domain $\Omega \subset \mathbb{C}$ has a logarithm.

Since $\Omega$ is simply connected, the logarithmic derivative of $f$ has a primitive $h$, i.e.

$$h'(z) = \frac{f'(z)}{f(z)}.$$

That means the function $f\cdot e^{-h}$ has derivative $\equiv 0$, so is constant. Choosing an appropriate constant, we obtain

$$f(z) = e^{h(z) + c}.$$

Since $\mathbb{C}$ is simply connected, that applies in particular to entire functions.

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