<caption><image_67>The image is a whiteboard with a hand holding a black marker. On the whiteboard, there are several equations written in black ink. The equations appear to be related to monotically increasing, as indicated by the logo on the top right corner of the image. 

The equations are written in a neat and organized manner, with some of them overlapping each other. The first equation is written in the form of a line, while the second equation is in the shape of a triangle. The third equation is a straight line, and the fourth equation is an arrow pointing to the right.

There are also several other equations written on the board, including a formula, a formula for the equation, and a formula that shows the relationship between the two variables. The formula is written on a piece of paper, and there is a logo of a company on the bottom right corner.</caption><transcript><audio_68>unique capital omega for every unique for every capital omega there's a unique small omega in spite of the fact that small omega is on a finite interval and capital omega is on an infinite interval we still have a one to one mapping this</transcript><caption><image_69>The image is a whiteboard with a mathematical equation written on it. The equation is written in black marker and is surrounded by various mathematical symbols and equations. The equations are written in a neat and organized manner, with some of them overlapping each other.

The equation reads "Monopoly Increasing" and is written above the equation. Below the equation, there is a formula that reads "8 = jian" which translates to "Jian" in English. The formula is written below the equation in a smaller font size. The image also has a logo of a company or organization on the top right corner. The background of the whiteboard is plain and unadorned.</caption><transcript><audio_70>it's one to one it's on two both ways so it's invertible and of course it is monotonically increasing as we wanted it to be well now about the left and the right half plane</transcript><caption><image_71>The image shows a large lecture hall with rows of students sitting at desks. The desks are arranged in neat rows and the students are facing the front of the room. The room has a high ceiling with recessed lighting and a projector screen in the background. The students are all wearing casual clothes and appear to be engaged in a discussion or lecture. The image appears to be taken from a high angle, looking down on the students.</caption><transcript><audio_72>so let's let's write the inverse mapping that's easier to deal with so we have s is 1 minus z inverse by 1 plus z inverse so of course i can write 1 plus z inverse times</transcript><transcript><audio_73>s is 1 minus z inverse so i have s plus s z inverse is 1 minus z inverse and that tells me that s plus 1 times z inverse</transcript><video><caption><image_74>The image shows a hand holding a black marker and writing on a whiteboard. The whiteboard has a logo on the top right corner that reads "CDEEP IT BOMBI" and a label on the bottom right corner. On the left side of the whiteboard, there is a mathematical equation written in black marker. The equation is written in a simple, hand-drawn style and appears to be a mathematical problem. The background is plain white.</caption><caption><image_75>The image is a photograph of a whiteboard with a mathematical equation written on it. The equation is written in black marker and is in the form of a square. Below the equation, there is a text that reads "7 = (1+8) = 1-8". On the right side of the image, there are two logos - one is the logo of the Indian government and the other is the emblem of the government of India. On the bottom right corner of the whiteboard, the text reads "CDEEP IT BOME 1".</caption></video><transcript><audio_76>and therefore z is one plus now we can say a lot about the real and imaginary parts let's</transcript><video><caption><image_77>The image shows a whiteboard with a mathematical equation written on it. The equation is written in black marker and is written on a white paper. The hand of a person is holding a pen and appears to be writing on the equation. The background of the image is blurred, but it seems like the person is focused on the task at hand. On the top right corner of the whiteboard, there is a logo of a company or organization, and on the bottom left corner, there are two logos, one for NPTEL and the other for CDEEP IT BOMB.</caption><caption><image_78>The image shows a whiteboard with a mathematical equation written on it. The equation is written in black marker and is written on a piece of paper with a black marker. The background of the image is blurred, but it appears to be a classroom or a classroom setting. On the right side of the whiteboard, there is a logo of a school or university with the text "CDEEP IT BOMB" and the logo of NPTEL. The image also has a watermark in the bottom right corner that reads "CDEEP IT BOME 1".</caption><caption><image_79>The image is a whiteboard with a mathematical equation written on it. The equation is written in black marker and is written on a piece of paper. The equations are written in a neat and organized manner. The background of the whiteboard is plain and there is a logo of a company on the top right corner. On the bottom right corner, there is text that reads "CDEEP H1 BOMB 1". The image appears to be a screenshot from a video or presentation.</caption></video><transcript><audio_80>let s p equal to sigma plus j omega 1 plus sigma plus j omega divided by 1 minus sigma minus</transcript><transcript><audio_81>and therefore we are interested in the mod in the magnitude of z it is the magnitude of z that concerns us let us see what happens to the magnitude</transcript><transcript><audio_82>the magnitude of z is of course the magnitude of the numerator numerator which is one plus in fact let's take magnitude of z squared it's the square of the magnitude of the numerator 1 plus sigma square plus omega square divided by the square</transcript> unique capital omega for every unique for every capital omega there's a unique small omega in spite of the fact that small omega is on a finite interval and capital omega is on an infinite interval we still have a one to one mapping this it's one to one it's on two both ways so it's invertible and of course it is monotonically increasing as we wanted it to be well now about the left and the right half plane so let's let's write the inverse mapping that's easier to deal with so we have s is 1 minus z inverse by 1 plus z inverse so of course i can write 1 plus z inverse times s is 1 minus z inverse so i have s plus s z inverse is 1 minus z inverse and that tells me that s plus 1 times z inverse and therefore z is one plus now we can say a lot about the real and imaginary parts let's let s p equal to sigma plus j omega 1 plus sigma plus j omega divided by 1 minus sigma minus and therefore we are interested in the mod in the magnitude of z it is the magnitude of z that concerns us let us see what happens to the magnitude the magnitude of z is of course the magnitude of the numerator numerator which is one plus in fact let's take magnitude of z squared it's the square of the magnitude of the numerator 1 plus sigma square plus omega square divided by the square
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