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:
©sideway ID: 190400013 Last Updated: 2019/4/13 Revision: 
Home (5) Computer Hardware (149) Software Application (187) Digitization (24) Numeric (19) Programming Web (554) CSS (SC) HTML Knowledge Base Common Color (SC) Html 401 Special (SC) OS (368) MS Windows Windows10 (SC) DeskTop (6) Knowledge Mathematics Formulas (8) Number Theory (206) Algebra (17) Trigonometry (18) Geometry (18) Calculus (67) Complex Analysis (13) Engineering Tables (8) Mechanical Mechanics (1) Rigid Bodies Statics (92) Dynamics (37) Fluid (5) Fluid Kinematics (5) Control Process Control (1) Acoustics (19) FiniteElement (2) Biology (1) Geography (1) 
Latest Updated Links

Copyright © 20002019 Sideway . All rights reserved Disclaimers last modified on 10 Feb 2019