<transcript><audio_62>takes you from capital omega going all over from minus to plus infinity minus pi goes to minus infinity 0 goes to 0 and plus pi goes to plus infinity and</transcript><caption><image_63>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 background of the whiteboard is plain white, and there is a logo on the top right corner that reads "Monotonically Increasing". The image appears to be a screenshot from a video or presentation.</caption><transcript><audio_64>all that we wanted out of the mapping between the unit circle and the imaginary axis has been</transcript><transcript><audio_65>as we move from minus pi to plus pi there is a also this mapping is one to one there is for every you see so i mean here you have j tan omega by 2 so essentially what you're saying is</transcript><transcript><audio_66>omega the capital omega is 10 small omega by 2 that's the mapping that you are and this is one to one that's the beauty of it you know for every small omega there's a</transcript> takes you from capital omega going all over from minus to plus infinity minus pi goes to minus infinity 0 goes to 0 and plus pi goes to plus infinity and all that we wanted out of the mapping between the unit circle and the imaginary axis has been as we move from minus pi to plus pi there is a also this mapping is one to one there is for every you see so i mean here you have j tan omega by 2 so essentially what you're saying is omega the capital omega is 10 small omega by 2 that's the mapping that you are and this is one to one that's the beauty of it you know for every small omega there's a
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