Complex Differentiabiity

Question 1

Differentiable

Answer 1

Suppose that S C_ (C and that f: S -> (C is a function. Then we say that f is differentiable at the point zo in S if
lim (z-> zo) (f(z) - f(zo) / (z-zo) exists

Question 2

Differentiable function (rewritten)

Answer 2

lim (w-> 0) f(zo +w) -f(zo) / w

Question 3

Differential of composition of functions

Answer 3

Suppose that zo e (C and the function f is differentiable at g (zo) and g is differentiable at zo. Then
the function f.g is differentiable at zo, and (f.g)’(zo) = f’(g(zo))g’(zo).

Question 4

Differentiable => continuous

Answer 4

Suppose that zo e (C and that the function f is differentiable at zo. Then
f(z) = f(zo) + f'(z)(z-zo) + E ( z, zo, f)
where lim (z-> zo) E (z, zo, f) / (z - zo) =0. 

Consequently f is continuous at zo.

Question 5

Relation between complex derivatives and partial derivatives

Answer 5

Suppose that S is an open subset of (C, that f: S -> (C is a function, that f(x+iy) = u(x,y) + i v(x,y), where u and v are real-valued functions of two real variables, and that f is differentiable at the interior point zo of S. Then the partial derivatives
du/dx (xo, yo) , du/dy (xo, yo) , dv/dx (xo, yo) , dv,dy (xo, yo)

Question 6

Cauchy-Riemann equations

Answer 6

du/dx (xo, yo) = dv/dy (xo, yo) and du/dy (xo, yo) = - dv/dx (xo , yo)

Question 7

Differential in terms of partial derivatives

Answer 7

f’(zo) = du/dx (xo, yo) + i dv/dx (xo, yo)

Question 8

Holomorphic / analytic

Answer 8

Suppose that f: % -> (C is a function, where % is an open subset of (C. If f is differentiable in the set %, (e.g. it is differentiable at every point of %), we say that f is holomorphic/analytic in %.
We write f e H(%)

Question 9

Entire

Answer 9

If % = (C

Question 10

Inverse function

Answer 10

Suppose that % and & are open subsets of (C, that f e H(%) and that f is one to one from % onto &. Then f has an inverse function, f^-1, from & to %.
We define f^-1(w) =z if f(z) =w.

Question 11

Branches

Answer 11

The different possible inverse function are called branches of the nth root function.

Question 12

Harmonic

Answer 12

Suppose that u : % -> |R is a function, where % is an open subset of |R^2, and that the partial derivatives
du/dx , du/dy , d^u/dx^2 , d^2u/dxdy , d^u/dydx and d^2u/dy^2 all exist and are continuous.

The we say that u is harmonic in % if d^2u/ dx^2 + d^2u/dy^2 = 0.

Question 13

Holomorphic functions => harmonic

Answer 13

Suppose that f e H(%), where % is an open subset of (C, and that
f(x + iy) = u( x, y) + iv(x , y) in %, where u and v are real-valued. Then u and v are harmonic functions.

Question 14

Harmonic conjugate

Answer 14

If % is a simply connected open set, and u : % -> |R is a harmonic function, there there exists a harmonic function v: % -> |R such that f, given by
f( x + iy) = u(x , y) + i v(x, y)
in % is holomorphic. Any two such functions differ by an additive constant.

The function v is called the harmonic conjugate of u.

Question 15

Applications of harmonic functions

Answer 15

• Electrical and gravitational potentials are harmonic
• Components of electrical and gravitational fields (in a fixed direction) are harmonic
• Measuring potential on the boundary of a body (We can find a harmonic function inside which, on the boundary,, is equal to a given function - Dirichlet problem)