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00:00:00 – 00:12:49
The video provides an in-depth explanation of Jordan's Lemma, a crucial theorem in the study of complex variables. The speaker introduces the concept by describing its conditions, including the analyticity of the function ( f(z) ) in the upper half-plane outside a certain radius and its behavior as it tends to zero at infinity. The lemma involves a contour integral over a semicircle, and its proof relies on Jordan's inequality, which bounds the integral of ( e^{-R sin theta} ) over specified intervals.
The instructor illustrates the lemma’s proof by graphing relevant functions, converting contour integrals into polar coordinates, and applying bounds derived from Jordan's inequality. They leverage an integral magnitude theorem for functions that are piecewise continuous over an interval to further simplify the expressions. By indicating how to handle the magnitudes and exponential terms, they demonstrate that the integral’s magnitude is ultimately bounded and approaches zero as the radius ( R ) goes to infinity.
In the concluding part, the speaker connects the lemma to the broader context of calculating improper integrals using the residue theorem, setting the stage for future discussions on practical applications. The video wraps up with acknowledgments and calls to action.
00:00:00
In this segment of the video, the instructor introduces Jordan’s Lemma, an important concept in complex variables. They clarify the pronunciation of the lemma’s name, stating it is French and pronounced “Jordan.” The instructor then sets up the necessary background by drawing the complex plane, marking the real and imaginary axes, a smaller circle with radius R₀ centered at the origin, and a larger semicircle with radius R also centered at the origin, where R is greater than R₀.
The lemma’s conditions are then listed: a function f(z) needs to be analytic (holomorphic) in the upper half-plane outside the circle of radius R₀. The semicircle, C_R, with radius R and equation z = R * exp(iθ) (where θ runs from 0 to π), must approach infinity. If f(z) is bounded on this semicircle and tends to 0 as R approaches infinity, then the contour integral of f(z) * exp(iaz) over C_R in the anti-clockwise direction also approaches 0 as R approaches infinity, with ‘a’ being a positive constant.
To prove this lemma, the instructor starts by graphing y = sin(θ) and y = (2θ)/π. This setup demonstrates that sin(θ) ≥ (2θ)/π for θ between 0 and π/2, leading to a crucial inequality for exponential functions that underpins the lemma’s proof.
00:03:00
In this part of the video, the speaker discusses the integral computation and Jordan’s inequality. They start by evaluating the ladder integral and proving that for R greater than 0, the integral from 0 to π/2 of e^(-R * sinθ) is less than or equal to π/(2R). They then extend this to the integral from 0 to π showing it is twice the previous integral, resulting in it being less than or equal to π/R. This is confirmed as Jordan’s inequality, which is used to prove Jordan’s lemma. The speaker proceeds by transforming the contour integral over a semicircle in the complex plane to an integral over an interval using the polar representation of complex numbers. They adjust the integral accordingly and emphasize changing the variable from Z to R * e^(iθ) and swapping DZ for dθ. Finally, they take the magnitude of the integral, hinting at a theorem related to their earlier work.
00:06:00
In this segment, the speaker discusses the application of an integral magnitude theorem to a complex function that is piecewise continuous over a specified interval. They explain that the function ( f(z) ) is analytic outside a smaller circle of radius ( R_0 ), allowing the theorem to be applied safely. Additionally, the magnitude of the product of two complex numbers is the product of their magnitudes. Applying this to the integral’s expression, the speaker explains splitting the magnitudes, including those for ( e^{itheta} ), which is simplified using Euler’s formula to show it equals 1. Further, they break down and simplify the exponential term within the integral, noting that for complex exponentials involving sine, the magnitude is manageable. Ultimately, they demonstrate how to replace and simplify parts of the integral, focusing on handling complex exponential expressions.
00:09:00
In this part of the video, the speaker explains how to apply Jordan’s inequality to an integral involving a complex function ( f(z) ). By representing the function in polar coordinates and leveraging the inequality ( f(z) leq M_R ), where ( M_R ) is a constant upper bound, the speaker demonstrates that the integral of ( f(z) ) times an exponential term is bounded. This is shown by moving constants outside the integral and applying Jordan’s inequality. By gradually simplifying the expression, they arrive at a key inequality which shows that the integral’s magnitude is ultimately bounded by ( M_R times pi / a ). They conclude that as ( R ) approaches infinity, the upper bound ( M_R ) approaches zero, hence the integral itself approaches zero, establishing the desired proof.
00:12:00
In this part of the video, the speaker explains the rationale that the magnitude of a contour integral approaching zero also means the contour integral itself approaches zero, thereby proving Jordan’s Lemma. Furthermore, the speaker indicates that in the next video, they will demonstrate how to use Jordan’s Lemma to calculate improper integrals using the residue theorem. The segment concludes with a thank you to patrons and an encouragement to like and subscribe.