Okay it's 6 steps and if you do it enough times it wil be down to only 5 steps.
Just so all of this actually makes sense, we will be using the following example:
Above equation is to be integrated along the circle|z|=R, starting and finishing on the route.
Before getting into the sateps lets make use we know the basics
An imaginary number is conventionally denoted as
and a function is denoted as
f(z) = u(x, y) + \iota v(x, y)
here `u` and `v` are functions of `x` and `y`, and that `i` thing is iota also known as root of -1.
Write the path in terms of only one variable, conventionally `t`. So, here we can write 'z' as:
from this we can directly compare and get `x` and `y`
x = R \cos t
y = R \sin t
That was all step 1 was.
Now we will get function which is in terms of 'u' and 'v' into terms of 'x' and 'y'.
From this we can take out values of 'u' and 'v' as
At this step substitute the earlier values of 'u' and 'v' we have in terms of 'x' and 'y', to have it in terms to 't'.
Now plug them into this equation you can do the following.
In the very likely case you forget the equation:
You can get it directly from this:
Just multiply those two brackets in the L.H.S and you'll be at the equation above it in 2 lines.