morale
发表于 2025-3-23 12:00:38
Limit of Functions,In Chapter III we dealt with limits of real sequences. These are real-valued functions whose domains are essentially ℤ. or ℤ.. In this chapter we treat limits of real-valued functions of a real variable whose domains are not necessarily confined to ℤ. or ℤ.. Of special interest are functions whose domains are intervals.
Nutrient
发表于 2025-3-23 15:16:26
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脱水
发表于 2025-3-23 20:33:05
Derivatives,Limits often arise from considering the derivative of a function at a point.
Charitable
发表于 2025-3-24 02:12:05
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铁砧
发表于 2025-3-24 05:01:25
,L’Hôpital’s Rule—Taylor’s Theorem,. (Cauchy’s Mean-Value Theorem). . [.,.], . (.; .) .′(.) ≠ 0 . ∈ (.; .), . (1) .(.) ≠ .(.); (2) . ∈ (.; .) ..(3) .(.) ≠ .(.), .. (1.1), .′(.) and .′(.) ..
Cacophonous
发表于 2025-3-24 08:04:51
The Complex Numbers. Trigonometric Sums. Infinite Products,In order to solve the equation.where ., ., . are real numbers and . ≠ 0, for . ∈ ℝ, we use the identity.obtained by “completing the square.” A real number . satisfying (1.1) must satisfy
BET
发表于 2025-3-24 13:24:14
Sequences and Series of Functions II,We consider power series . and . with respective radii of convergence . and . and write . = min{., .}. We also assume that . > 0 so that . ⩾ . > 0 and . ⩾ . > 0.
Free-Radical
发表于 2025-3-24 16:53:52
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我不怕牺牲
发表于 2025-3-24 19:33:00
The Riemann Integral II,We now consider the legitimacy of passing to the limit under the integral sign. If the sequence 〈.〉 of .-integrable functions converges to a limit . on an interval [., .] does it necessarily follow that
异端邪说下
发表于 2025-3-24 23:42:48
Improper Integrals. Elliptic Integrals and Functions,When . is .-integrable over [., .] then its indefinite integral ., defined as.is continuous on [.,.] (Theorem XIII.6.3). Hence,