Integrate Complex Conditional with airSlate SignNow

Integrate complex required

here we're going to explore some of the strangeness that is complex integration what do we even mean by a complex integral like is the integral from alpha to beta of f of z dz well-defined where alpha and beta are complex numbers and i really ask if it's well defined because if you're thinking about the complex plane and two numbers alpha and beta in the complex plane then there are infinitely many paths that you can take from alpha to beta so those are two such paths we could have another path kind of like squiggling in the middle like that if this type of thing is well defined where all we're doing is talking about the starting point and the ending point then that must say something about taking the integral about a different path okay so how are we going to do this exploration well we're going to set up two paths that go from zero to the point one plus i that's a complex number and then we'll integrate two functions over those paths so what will the paths in question be well maybe the most obvious one first which is the line segment starting at 0 and ending at 1 plus i so notice that we can parametrize that by this thing i'm calling gamma 1 which is 1 plus i times t so at t equals 0 we're at the origin and at t equals one we're at the point one plus i then for our second path we'll actually take a kind of two piece path given by gamma two and gamma three so gamma two of t is just t so as t goes between 0 and 1 that takes us from the origin to this point which is 1 unit along the positive real axis and then gamma 3 is 1 plus i t so as t goes between 0 and 1 that takes us from this point 1 to this point up here 1 plus i so in the end the combination of gamma 2 and gamma 3 represent a different path from the origin to the point 1 plus i and if you want to see this more in depth along with some general proofs i've got a whole course on my second channel of complex analysis and so that channel is called math major okay so let's first do the integral over each path of the function z dz okay so let's see our first path is gamma 1 and so that means z will be equal to gamma 1 and our parameter like i said goes between 0 and 1 so that gives me the integral from 0 to 1 of gamma 1 of t so that's going to be 1 plus i times t times dz but dz will just be gamma 1 prime dt so the derivative of this with respect to t is just 1 plus i and then we have a dt...