Question

Assume that a>0,μ>0a>0,μ>0 and look at the complex function f(z)≐exp(−aμ2−2z−−−−−−√−μ).f(z)≐exp⁡(−aμ2−2z−μ). From what I understand we need a branch of the logarithm for the square root to be defined? So we can define this function for R(z)μ2/2x>μ2/2?​

Answer 1

Answer:

Definition – Complex-Differentiability & Derivative.

‌ Let f:A⊂C→C.f:A⊂C→C. The function ff is complex-differentiable at an interior point zz of AA if the derivative of ff at z,z, defined as the limit of the difference quotientf′(z)=limh→0f(z+h)−f(z)hf′(z)=limh→0f(z+h)−f(z)hexists in C.C.

Remark – Why Interior Points?

‌ The point zz is an interior point of AA if∃r>0,∀h∈C,|h|<r→z+h∈A.∃r>0,∀h∈C,|h|<r→z+h∈A.In the definition above, this assumption ensures that f(z+h)f(z+h) – and therefore the difference quotient – are well defined when |h||h| is (nonzero and) small enough. Therefore, the derivative of ff at zz is defined as the limit in “all directions at once” of the difference quotient of ff at z.z. To question the existence of the derivative of f:A⊂C→Cf:A⊂C→C at every point of its domain, we therefore require that every point of AA is an interior point, or in other words, that AA is open.

Definition – Holomorphic Function.

‌ Let ΩΩ be an open subset of C.C. A function f:Ω→Cf:Ω→C is complex-differentiable – or holomorphic – in ΩΩ if it is complex-differentiable at every point z∈

Step-by-step explanation:

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