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00:00:00 – 00:12:49
The video primarily explores Jordan's Lemma, a significant concept in complex analysis, and provides a step-by-step proof of the lemma. The instructor starts by explaining the background and importance of Jordan's Lemma, discussing its application to contour integrals in the complex plane. Key to the discussion is the understanding that if the function ( f(z) ) is analytic in the upper half-plane and diminishes to zero as the radius ( R ) approaches infinity, then the contour integral of ( f(z) ) over a semicircle also tends towards zero.
The explanation includes demonstrating auxiliary concepts like Jordan's Inequality and exploiting the symmetry of the sine function to simplify the integral calculations. The instructor illustrates techniques to manage the integral of a complex function using polar coordinates and properties of complex numbers, notably leveraging the magnitude component of complex exponentials and Euler's formula. They meticulously walk through transforming complex contour integrals to integrals over intervals, ultimately illustrating that the integral is simplified and bounded by constants that render the result zero as ( R ) increases indefinitely.
The culmination of the video is the formal proof that the magnitude of the integral over the contour ( C_R ) approaches zero, thus proving Jordan's Lemma. The speaker concludes by confirming that the actual contour integral also approaches zero when the magnitude does, reinforcing the validity of the lemma. The lesson wraps up with a teaser for the next topic, which involves applying Jordan's Lemma to evaluate improper integrals with the residue theorem.
00:00:00
In this part of the video, the instructor discusses Jordan’s Lemma, a concept in complex variables. The instructor begins by clarifying the correct pronunciation of “Jordan” as it is a French name. They draw the complex plane with a real and imaginary axis, a smaller circle with radius ( R_0 ), and a larger semicircle with radius ( R ). The semicircle is labeled ( C_R ), with ( R ) greater than ( R_0 ).
They state that if a function ( f(z) ) is analytic in the upper half-plane outside the smaller circle and approaches zero as ( R ) approaches infinity, then the contour integral of ( f(z) ) times the exponential of ( iaz ) over the semicircle approaches zero as ( R ) approaches infinity, where ( a ) is a positive constant.
The instructor plans to prove the lemma and starts by graphing the functions ( y = sin theta ) and ( y = 2theta/pi ). They show that for ( theta ) between 0 and (pi/2), (sin theta ) is greater than or equal to ( 2theta/pi ). This implies that the negative exponential of ( R sin theta ) is less than or equal to the negative exponential of ( R cdot 2theta/pi ), with ( R ) being a positive constant.
00:03:00
In this segment of the video, the speaker explains how to compute an integral involving exponential and sine functions, demonstrating that the integral from 0 to π by 2 of e^(-R * sin(θ)) is less than or equal to π/2R, where R is a positive number. The symmetry of the sine function allows the extension of this result to an integral from 0 to π. This leads to the introduction of Jordan’s Inequality, which is used to prove Jordan’s Lemma. The speaker then transitions to proving Jordan’s Lemma by transforming a complex contour integral over a semicircle into an integral over an interval using the polar representation of complex numbers. Finally, they express the contour integral in terms of θ and discuss adjusting the limits and integrating over θ.
00:06:00
In this part of the video, the speaker explains the application of an integral magnitude theorem to a complex function that is piecewise continuous over a given interval. The function in question is analytic outside a smaller circle of radius (R_0). The speaker reminds that the magnitude of the integral of a function is less than or equal to the integral of the function’s magnitude and applies this concept to split up the magnitudes inside the integral.
They then use properties of complex numbers, particularly the magnitude of the product of two complex numbers being the product of their magnitudes. The specifics include splitting the magnitude into components and simplifying ( e^{itheta} ) using Euler’s formula, which yields a magnitude of 1. For the exponential term with a complex exponent, they expand it using trigonometric identities from Euler’s formula, focus on handling the real part separately (resulting in its magnitude being just the real number), and disregard components with a magnitude of 1.
Finally, the speaker concludes that the exponential component ( e^{(i cdot a cdot R)} cdot e^{(i theta)} ) simplifies to ( e^{-aRsin(theta)} ). They then proceed to replace this form back into the integral they were evaluating, readying for further calculations.
00:09:00
In this segment of the video, the speaker explains how to handle the integration of a complex function f(z) using the polar representation of complex numbers. They substitute the function f with its upper limit M sub R in the integral and manage to simplify this integral by recognizing that both M sub R and R are constants. The integral of the exponential term is shown to resemble Jordan’s inequality, leading to the simplification of the expression to M sub R times π over a. By applying a series of equalities and inequalities, they demonstrate through the transitive property that the magnitude of the integral over the contour C R is less than or equal to this simplified term. The final step involves taking the limit as R approaches infinity, showing that the limit of the integral is less than or equal to zero, concluding that the magnitude of a complex number must ultimately satisfy the equality due to its non-negative property.
00:12:00
In this part of the video, the speaker confirms that the limit of the magnitude of the contour integral is zero, leading to the conclusion that the limit of the contour integral itself must also be zero. This is because as the magnitude of the complex number approaches zero, the complex number itself approaches zero and gets closer to the origin. The speaker concludes by indicating that Jordan’s Lemma has been proven. In the next video, the speaker plans to demonstrate how to apply Jordan’s Lemma to calculate improper integrals using the residue theorem. The speaker also thanks patrons for their support and encourages viewers to like and subscribe.