Complex Analysis
Complex Analytic Function
Complex Logarithm Function
Draft for Information Only ContentComplex Function source/reference: Complex FunctionComplex Logarithm FunctionGiven 𝑧∈ℂ\{0},, find 𝑤∈ℂ such that ℯ^{𝑤}=𝑧 𝑧=𝑧ℯ^{𝑖𝜃}, then ℯ^{𝑤}=𝑧ℯ^{𝑖𝜃} Next, write 𝑤=𝑢+𝑖𝑣. Then ℯ^{𝑢}ℯ^{𝑖𝑣}=𝑧ℯ^{𝑖𝜃} Thus ℯ^{𝑢}=𝑧 and ℯ^{𝑖𝑣}=ℯ^{𝑖𝜃}, so 𝑢= Therefore, 𝑤= By definition. For 𝑧≠0,
and
Examples
Continuity of the Logarithm FunctionFor the logarithm function,
Derivative of Logarithm FunctionBy Fact. The principal branch of logarithm, The derivative:
ℯ^{Logz}=z
More General TheoremBy theorem. Suppose that 𝑓:𝑈→ℂ is an analytic function and there exists a continuous function 𝑔:𝐷→𝑈 from some domain 𝐷⊂ℂ into 𝑈 such that 𝑓(𝑔(z))=z for all z∈𝐷. Then 𝑔 is analytic in 𝐷, and 𝑔′(z)=
Application 1Let 𝑓:ℂ→ℂ, 𝑓(z)=z^{2}. Then 𝑓′(z)=2z. Let 𝑔:ℂ\(−∞,0]→ℂ, 𝑔(z)=√z be the principal branch of the square root (=√z⋅ℯ^{𝑖Argz2}). Then
Application 2Let 𝑓:ℂ→ℂ, 𝑓(z)=z^{2}. Then 𝑓′(z)=2z. Let ℎ:ℂ\[0,∞)→ℂ, ℎ(z)=
TerminologyLet 𝑓:𝑈→𝑉 be a function.
Examples:
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