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Integrate complex initials

integration is a fundamental operation in calculus and is probably one of the first things you learn about in a course on the subject one great thing about real definite integrals is that they have a very nice geometric interpretation that is the signed area between a curve and the x-axis in r2 which any math student will recognize but what about an integral over the complex numbers is there any geometric interpretation here in this video i hope to demystify complex integration a little bit for you the first idea to go over is that of a vector field in the complex plane we can represent any complex number as a vector writing a complex number in exponential form allows us to quickly determine the angle the complex number makes with the real axis in the plane as well as the absolute value of the complex number now suppose we have a function f of z for example say f of z is equal to one over z if we take any complex number z as our input we get a new complex number back and we can again think of this output number as a vector evaluating f of z at one plus i we get one half times one minus i a vector pointing into the fourth quadrant of the complex plane now imagine translating the tail of this output vector to our input point 1 plus i this is the vector we want to see in our vector field if we then do this with every output vector for our function we can look directly at our input point on the plane and see that the vector emanating from it captures the phase and the modulus of f z at that point the idea of two-dimensional work and flux is pretty straightforward suppose we have two vectors a and b we can define work simply as the dot product of these two vectors which is the absolute value of a multiplied with the absolute value of b all multiplied by the cosine of the angle theta between these two vectors when the angle between a and b becomes smaller cosine of theta approaches 1 and the absolute value of the dot product in turn becomes larger if the angle between a and b gets closer to 90 degrees cosine of theta approaches zero and thus the absolute value of the dot product becomes smaller flux can be defined as the dot product between vector b and a 90 degree rotation of vector a in other words the absolute value of a multiplied with the absolute value of b all multiplied by negative sine of the angle between these two vectors when the angle between a and b is smaller sine of theta is smaller and in turn the absolute value of the flux becomes smaller if the angle between a and b approaches 90 degrees sine of theta approaches one and thus the flux becomes larger...