Integrate Initial Fact with airSlate SignNow

Integrate initial fact

welcome back to our second lecture in the fifth week of our course analysis of a complex kind today we'll learn more about complex integration we'll look at some examples and we'll learn some first facts remember this is how we define the complex path integral given a smooth curve gamma and a complex valued function f that is defined on gamma we define the integral over gamma f of c dz to be the integral from a to b f of gamma of t times gamma prime of t dt so this second integral can be broken up into its real and imaginary parts and then integrated according to the rules of calculus let's look at some more examples suppose gamma of t is given by 1 minus t times 1 minus i where t runs from 0 to 1. let's get a quick idea of what this path looks like when t is equal to zero gamma of t equals one and when t is equal to one gamma of one is equal to one minus 1 minus i in other words i so that is gamma of 1. and what happens to the path in between in between there's a linear relationship between x of t and y of t if you write gamma of t as x of t plus i y of t then the real part is 1 minus t and the imaginary part is simply t so y is equal to t x is equal to one minus t they're linearly related so we just get this line segment from one to i let's see what the integral does the integral over gamma f of z dz by definition is the integral from zero to one these are the bounds for the t values of the function f the function f of z is given by the real part of z so we have to take the real part of gamma of t and multiply that by gamma prime of t what is gamma prime of t gamma prime of t is what the derivative of one is zero derivative of minus t times one minus i is minus 1 minus i so remember the path integral integral over gamma f of z dz is defined to be the integral from a to b f of gamma of t gamma prime of t dt that's what we're using right here so the integral over gamma f of c dz is the integral from zero to one f is the function that takes the real part of whatever is put into it we're putting gamma of t into it that's right here and then we multiply by gamma prime of t so this is the integral we have to evaluate let me clear the screen here to evaluate this integral we need to find the real part of 1 minus t times 1 minus i but the real part is everything that's real in...