Understanding the Lemma for L'Hospital's Rule: Analyzing the Proof

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SUMMARY

The discussion centers on the lemma used in proving L'Hospital's Rule, specifically the expression f(z) = f(z0) + f'(z0)(z - z0) + η(z - z0), where η approaches 0 as z approaches z0. The lemma is derived from the definition of the derivative and the properties of analytic functions. A participant clarifies a misunderstanding regarding the arrangement of terms in the equation, confirming that the original expression is indeed correct despite initial confusion over its formulation.

PREREQUISITES
  • Understanding of complex analysis, particularly analytic functions.
  • Familiarity with the definition of derivatives in complex functions.
  • Knowledge of L'Hospital's Rule and its applications.
  • Basic algebraic manipulation skills to rearrange equations.
NEXT STEPS
  • Study the proof of L'Hospital's Rule in detail.
  • Explore the properties of analytic functions in complex analysis.
  • Learn about Taylor series expansions and their relation to derivatives.
  • Investigate common pitfalls in understanding limits and continuity in complex functions.
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Students of complex analysis, mathematicians interested in calculus, and educators teaching L'Hospital's Rule and its foundational lemmas.

demonelite123
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Suppose f(z) is analytic in a region R including the point z0. Prove that f(z) = f(z0) + f'(z0)(z-z0) + η(z-z0) where η ~> 0 as z ~> z0.

this is actually a lemma my book proves first before actually proving L'Hospital's rule. I understood how they used the lemma to prove the rule but i don't really understand the logic in proving this lemma. my book did:

Let [f(z) - f(z0)]/(z-z0) - f'(z0) = η so that f(z) = f(z0) + f'(z0)(z - z0) = η(z-z0).
Then, since f(z) is analytic at z0, we have as required:
lim (z ~> z0) of η = lim (z ~> z0) of [f(z) - f(z0)]/(z-z0) - f'(z0) = f'(z0) - f'(z0) = 0.

i don't understand how f(z) = f(z0) + f'(z0)(z - z0) = η(z-z0). shouldn't it be f(z) = η(z-z0) + f'(z0)(z - z0) + f(z0) since they let [f(z) - f(z0)]/(z-z0) - f'(z0) = η?
 
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demonelite123 said:
i don't understand how f(z) = f(z0) + f'(z0)(z - z0) = η(z-z0). shouldn't it be f(z) = η(z-z0) + f'(z0)(z - z0) + f(z0) since they let [f(z) - f(z0)]/(z-z0) - f'(z0) = η?

Hi demonelite123! :smile:

It's just a typo :rolleyes: … + and = are the same key on most keyboards! :wink:
 
oh no wonder. thanks!
 

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