11.10.2024 Лекция 5. Первообразная и теоремы Коши

Continuing the Discussion on Integrals and Antiderivatives

Review of Previous Concepts

• The session begins with a prompt to continue from where they left off, inviting questions about previous topics discussed.

• The discussion revisits properties of integrals and introduces the concept of antiderivatives, hinting at a connection to Newton-Leibniz formula.

Introduction to Antiderivatives

• An antiderivative F for a function f is defined on an open interval D , emphasizing that if the derivative of F equals f , then this relationship holds true within that interval.

• A theorem analogous to the Newton-Leibniz formula is proposed, suggesting that if a function has an antiderivative, certain properties can be derived.

Conditions for Continuity and Differentiability

• The speaker emphasizes continuity conditions for functions when discussing their integrability and existence of antiderivatives.

• It is noted that both continuity and differentiability are crucial in establishing relationships between functions and their integrals.

Piecewise Smooth Paths

• The conversation shifts towards piecewise smooth paths connecting points P and Q , highlighting how these paths relate to integration.

• A question arises regarding the value of an integral along such paths, leading to insights about evaluating integrals based on endpoints.

Definition and Existence of Integrals

• The definition of an integral over a curve is revisited, clarifying its dependence on parameterization.

• Continuous functions are deemed necessary for defining integrals properly; thus, ensuring the existence of these integrals becomes paramount.

Implications of Differentiability

• There’s a discussion about differentiable functions being holomorphic in open sets, which implies they possess infinite derivatives under certain conditions.

• The speaker suggests that while continuity may seem excessive as a requirement for some results, it plays a critical role in understanding the behavior of functions involved in integration.

Conclusion: Holomorphic Functions

Understanding Closed Paths and Integrals

Theorem Conditions and Closed Paths

• Discussion on the implications of a closed path (gamma) in relation to integrals, emphasizing that if gamma is closed, certain conditions apply.

• Clarification that a closed path means the start and end points coincide, with no additional conditions affecting this definition.

• It is noted that the integral of a continuous function with an antiderivative over a closed path equals zero, highlighting the significance of continuity.

Example of Integration Over Closed Paths

• Reference to an example involving integration of Z^N over a circular path (gamma), specifically when gamma represents some circle in the complex plane.

• The speaker recalls previous discussions about integrating functions like Z^N , questioning whether they had previously established results regarding such integrals.

Key Results from Integration

• Emphasis on remembering specific constants derived from these integrations, particularly noting that 2pi will frequently appear in future discussions.

• A question posed about whether there exists an antiderivative for 1/Z across the complex plane excluding zero, hinting at logarithmic relationships.

Challenges with Antiderivatives

• Exploration of what would happen if an antiderivative existed for 1/Z ; it would imply that integrals around any closed contour should equal zero, which leads to contradictions based on prior examples.

• Discussion on how other functions may not have straightforward antiderivatives but do not contradict existing results; highlights complexities in determining these properties.

Logarithm as a Multi-Valued Function

• Examination of why logarithms cannot serve as single-valued antiderivatives due to their multi-valued nature and periodicity issues related to angles less than 2pi .

• Explanation that logarithmic functions require careful selection of branches or sectors to maintain consistency and avoid ambiguity during integration.

Implications for Contour Integrals

• Insight into how contours encircling singularities (like zero for 1/Z ) yield non-zero integrals while those avoiding such points result in zero contributions.

Understanding Complex Functions and Cauchy's Theorem

Introduction to Multivalued Functions

• Discussion on multivalued functions, particularly when the angle exceeds 2pi, leading to ambiguity in values.

• Explanation of closed contours that do not enclose points, specifically relating to logarithmic functions and their derivatives.

Logarithmic Functions and Their Properties

• Emphasis on the complexity of logarithmic functions, especially when crossing zero, which can lead to different branches of values.

• Introduction of Cauchy's theorem as a foundational concept in complex variables, indicating its importance for future discussions.

Defining Triangles in Complex Analysis

• Agreement on understanding triangles without strict definitions; acknowledgment that they are closed figures with three angles.

• Clarification that a triangle will be treated as a standard figure in the plane for analysis purposes.

Cauchy's Integral Theorem

• Presentation of Cauchy’s first theorem regarding holomorphic functions within a triangle completely contained in a domain D.

• Definition of boundaries for triangles and how these relate to integrals over holomorphic functions.

Implications of Holomorphic Integrals

• Discussion about integrating holomorphic functions over triangular boundaries and implications for general closed contours.

• Exploration of intuitive reasoning behind why integrals over holomorphic functions yield specific results based on their properties.

Understanding Function Behavior Near Points

• Examination of function behavior near specific points (e.g., z_0) and how this relates to integral calculations.

• Insight into linear approximations around points and the significance of continuity in determining integral outcomes.

Conclusion: The Nature of Holomorphic Functions

• Summary statement about the relationship between continuous functions having primitives (antiderivatives), leading towards further exploration into Cauchy’s theorem implications.

Analysis of Triangles and Integrals

Understanding the Concept of Small Values in Analysis

• The speaker discusses the behavior of a small value as Z approaches zero, questioning why certain terms can be disregarded when they are also small.

• A standard proof in analysis is introduced, focusing on approaching a point Z with zero. The speaker emphasizes the importance of understanding this approach through geometric representation.

Geometric Representation and Triangle Division

• The triangle Δ is divided into four smaller triangles using three midlines, illustrating how integration over curves relates to these subdivisions.

• The speaker establishes a convention for positive orientation in triangles, indicating that all internal triangles will also be treated positively during integration.

Integral Relationships and Summation

• An equation is proposed where the integral over the original triangle equals the sum of integrals over its subdivisions, highlighting logical consistency in this relationship.

• The directionality of traversal along triangle sides affects integral values; opposing directions lead to cancellation in integrals.

Observations on Integrals and Non-Zero Assumptions

• A hypothesis is presented regarding non-zero integrals; if an integral's absolute value is not zero, it leads to further implications about individual integrals within subdivisions.

• The speaker notes that the absolute value of a sum does not exceed the sum of absolute values, prompting questions about individual integral evaluations.

Recursive Process and Nested Triangles

• It’s suggested that at least one integral among those evaluated must meet specific criteria based on previous assumptions about their magnitudes.

• Continuing with nested triangles (Δ1, Δ2,...), if one integral meets a threshold condition, it implies similar conditions for subsequent divisions leading to increasingly smaller areas while maintaining proportional relationships.

Understanding Perimeter and Points in Geometry

Defining Perimeter with Delta Notation

• The speaker introduces a notation for perimeter, denoting it as "per Delta N," which is defined as textper Delta = 2^N . This notation is presented as a clever way to express the concept.

Exploring Point Z with Zero

• The discussion shifts to point Z, referred to as "Z with zero." It is suggested that this point can be considered as the intersection of all triangles involved in the discussion.

Intersection of Triangles

• The speaker emphasizes that point Z belongs to every triangle by examining the intersection of triangles from zero to infinity. This leads to questions about analogous theorems related to nested segments in higher dimensions.

Properties of Closed Triangles

• A theorem regarding closed triangles is mentioned, indicating that if these triangles are shrinking (with diameters approaching zero), there exists a unique point within their intersection.

Role of Analyticity at Point Z

Understanding Integral Properties and Theorems

Key Insights on Integrals

• The discussion begins with the explanation of a derivative, leading to the conclusion that only the integral remains after certain terms are simplified. The focus is on how small terms can be disregarded in this context.

• A theorem is introduced stating that the modulus of an integral does not exceed the integral of its modulus, reinforcing a fundamental property of integrals.

• The speaker applies this theorem, indicating that if a function f(z) is bounded by its supremum over a curve, it can be used to evaluate integrals effectively.

• The length of Delta N , defined as half the perimeter divided by 2^N , is discussed. This establishes a relationship between geometric properties and integrals.

• An inequality involving supremums and products is presented, emphasizing that for all points within a triangle, certain bounds hold true based on previous definitions.

Geometric Considerations in Integration

• The distance from point z_0 to any point within a triangle is evaluated using geometric principles, suggesting that distances are constrained by the triangle's sides.

• It’s noted that any two points within a triangle will have distances less than or equal to the longest side of the triangle, which relates back to perimeter considerations.

• Further elaboration on perimeter shows how it influences distance calculations among points in geometric shapes like triangles.

• A derived expression indicates relationships between constants and perimeters squared; however, it's acknowledged that while this may seem trivial, accuracy in these expressions is crucial for further conclusions.

Convergence and Contradictions

• A critical inequality emerges showing that as one term approaches zero (denoted as M ), contradictions arise if assumptions about non-zero values hold true.

• This contradiction leads to significant implications regarding integrals being zero under specific conditions when approaching limits—highlighting foundational concepts in analysis.

Existence of Antiderivatives

• Questions arise regarding proofs related to antiderivatives for holomorphic functions. It’s suggested that understanding these concepts requires familiarity with prior mathematical theories.

• Introduction to convex sets occurs here; defining them through their properties helps establish criteria for functions having antiderivatives within those sets.

Convexity and Holomorphic Functions

• Examples illustrate what constitutes convex sets versus non-convex ones. Understanding these distinctions aids in grasping more complex mathematical structures later discussed.

• The existence of an antiderivative for holomorphic functions is asserted; parallels drawn with continuous functions suggest similar methodologies apply here for establishing such properties.

Understanding Derivatives and Integrals in Convex Sets

Definition of Derivative

• The speaker discusses the derivative of a function f at point Z , emphasizing the need to demonstrate that it is an antiderivative.

• A convex set D is introduced, with a focus on its properties and how they relate to the calculation of derivatives.

Geometric Interpretation

• The speaker illustrates a triangle formed by points Z_0 , A , and another point in relation to the convex set, indicating that this triangle lies entirely within D .

• The integral over the boundary of this triangle is discussed, suggesting that it equals zero due to properties of integrals over closed paths.

Integral Calculation

• The process for calculating the integral from point Z_0 to point A_pH is outlined, including considerations for additive properties of integrals.

• There’s a discussion about rearranging terms in the integral expression to facilitate simplification.

Antiderivative Properties

• The speaker notes that integrating from one point to another can be manipulated by changing limits and signs, leading towards establishing relationships between values at different points.

• An important relationship emerges: the difference between function values at two points relates directly to their respective integrals.

Continuity and Holomorphic Functions

• The continuity of holomorphic functions is emphasized; as these functions are continuous, their behavior near specific points can be analyzed effectively.

• It’s noted that if a function has an antiderivative, then certain integral properties hold true across closed contours.

Conclusion on Antiderivatives

• The conclusion drawn is that if a holomorphic function has an antiderivative, then its integral around any closed contour will equal zero. This leads into discussing Cauchy's theorem.

Theorems in Complex Analysis

Introduction to Theorem 2

• The speaker introduces a theorem related to triangles, indicating its relevance in proving certain properties.

• The discussion shifts to convex sets, emphasizing the importance of closed paths within a domain D .

Observations on Integrals

• First observation: If gamma is a closed path in D , then the integral of function f over this path equals zero.

• Second observation: For two paths gamma_1 and gamma_2 connecting points P and Q , both lying in D, their integrals are equal.

Implications of Holomorphic Functions

• It is noted that if two different paths connect the same points, their integrals will yield the same result.

• The speaker contemplates whether these results need formal proof or if they are intuitively clear based on the existence of primitives for holomorphic functions.

Moving Beyond Convexity

• A desire to move away from convexity is expressed; however, it’s acknowledged that convexity simplifies many arguments.

• An example involving a star-shaped region is presented, questioning whether an integral around such a contour would still yield zero.

Exploring Non-convex Domains

• The speaker discusses how non-convex regions can complicate matters but suggests using techniques applicable within convex parts.

• Introduction of Cauchy's third theorem, which requires careful formulation due to its complexity.

Conditions for Cauchy’s Third Theorem

• Assumes function F is holomorphic in an open set D; closed paths must not intersect each other.

• Focuses on scenarios where one path lies inside another while ensuring that the area between them also belongs to domain D.

Integral Equality Between Paths

• States that under these conditions, the integrals over both paths remain equal regardless of their shapes or positions as long as they meet specified criteria.

Conclusion on Path Independence

• Emphasizes that even without knowledge about convexity, if boundaries lie within domain D, integrals can still be evaluated as zero under certain conditions.

Understanding Contours and Integrals in Convex Sets

Breaking Down Contours

• The discussion begins with two contours, gamma 1 and gamma 2, emphasizing the ability to break down these contours into smaller segments.

• A large set D is introduced, which can be partitioned into convex parts. This segmentation aids in analyzing the properties of integrals over these contours.

Positive Direction Integration

• Both gamma 1 and gamma 2 are traversed in a positive direction. The choice of convex parts is crucial for ensuring that the integration process remains valid.

• It is noted that when integrating over these convex sections, the integral evaluates to zero due to the closed nature of paths within a convex area.

Summation of Integrals

• The sum of integrals across all segments results in zero. This conclusion stems from the fact that each segment's integral contributes nothing when enclosed within a convex region.

• The total integral across all pieces equates to the integrals along gamma 1 and gamma 2 but taken in opposite directions.

Conclusion on Integral Equality

• Ultimately, it follows that the integral along gamma 1 equals that along gamma 2 when considered in their respective correct orientations.