Complex methods Integration Flashcards
Express sin, cos and e^x as a summation
Determine if the following functions have branch points:
f(z) = z^2
f(z) = ln(z)
f(z) = z^(1/2)
What is a branch point?
A branch point is a point in the complex plane in which a function is multivalued.
Both the square root and log functions are multivalued about the origin.
How do you check whether a function has a branch point at infinity?
Try this for ln z and z^1/2
Change of variables –> z=1/w
What is the difference between an algebraic and a trancendental/logarathmic branch poin?
What is a branch cut and why do we do them?
A branch cut is a line which joins branch points and cuts them out of the complex plane.
This makes the function everywhere else in the plane smooth and continuous.
What is the derivative of a complex function?
When is a function analytic?
Is f(z) = z* analytic?
Function is analytic when df/dz = constant, i.e. we get the same finite answer when we choose that t approahes zero from any direction in the complex plane.
Derive the Cauchy-Rienmann conditions
Hint: Start with f(z) = u(x,y) + iv(x,y)
Using C-R conditions:
Is f(z) = z^2 analytic?
Is f(z) = z* anayltic?
Using C-R conditions: Reconstruct the following function f(z).
f(z) is some analytic function where u(x,y) = x^2-y^2
What is Cauchy’s theorem?
Cauchy’s theorem: If a function f(z) is analytic in a simply connected region R, and C is a contour in R which is simple and closed then the contour integral of f(x) = 0/.
Prove Cauchy’s theorem
Deforming contours- Integrating along ANY contour between two points A and B will have the same value/result for any analytic function f(z). df/dz = const
Proof shown in the image using Cauchy’s theorem.
What is a regular point, a singular point and an isolated singular point?
Another way to test if a function is analytic is to let t be purely real or imaginary.
Using this method, test if f(z) = z* is analytic
Where are the singular points for these functions?
What is Cauchy’s integral formula?
Prove Cauchy’s integral formula
Needed to put answer on both sides so don’t look at image.
CIF and its derivatives.
Using Cauchy’s integral formula, find CIFs derivatives (just do up to first derivative).
Expand
f(z) = e^(1/z) about z=0
f(z) = z/(z-a) about z= a.
What is a residue?
