00:01
Hi, in this video we're going to use the maximum modulus principle to find the maximum value of the modulus of these two functions.
00:12
So we're looking at the disk, so like our domain is the disk, absolute value of z is less than equal to 1.
00:24
So we're just looking at the unit disk.
00:31
So then first we're looking at f of z is equal to e to the z squared plus 1.
00:38
And i'm going to write that as e times e to the z squared.
00:45
And so what the maximum modulus principle says is that since this is not a constant function, the maximum value has to occur on the boundary of our disk.
00:54
So the max has to occur when the modulus of z is actually equal to one.
01:02
And then the question is which one of these points on the disk does f of z have a max? so if we're looking at modulus of z is equal to 1, then we can write z as e to the i theta for some theta, right? the r is 1.
01:25
So that means f of z is e times e to the i squared, which is negative 1.
01:33
Maybe i'll just write that.
01:35
I squared, theta squared.
01:40
So this is e times e to the negative theta squared.
01:54
Hi.
01:56
And so we know that theta squared is always going to be some non -negative number.
02:02
So since we have e to the negative theta squared, when is this maximized? well, we know if theta becomes anything far from zero, theta squared will become, will be some positive number, and then e to the negative theta squared will be decreasing...