MAS381 Mathematics III

发表于 2021-09-27 分类于 academic material

week1

complex value

f(z)=f(x+iy)=u(x,y)+iv(x,y)
Re(f)=x=u(x,y)
Im(f)=y=v(x,y)

harmonic function:

By definition a function is called harmonic if it is at least twice differentiable and it satisfies the well-known Laplace equation.

Euler’s formular:

cause

week 2

complex differentiation

A complex function f(x) is differentiable at a point z if and only if the limiting ratio quotient exists.

Cauchy-Riemann equation:

a way to defferentiate a complex funtion

where u=real part and v=imagin part, and a function has a complex derivative f’(z) if and only if its real and imaginary parts are continuosly differentiable and satisfy the Cauchy-Riemann equation:

which is called the conjugate.

also, Cauchy-Riemann equation can find harmonic function as:

analytical functions

A complex function f(z) is called analytic at a point z0 if it has a power series expansion

we can use ratio test to find wheter a complex series is convergent or not by finding the limitation value of n-th term

if L<1, convergent series
if L>1, divergent series
if L=1, the test is inconclusion.

for complex power series, the sequence Cn converges with a limit L. If L=0, the power series converges for all z. If L not equal 0, then

There are four and only four possible types of singularities of a complex function:

pole of order n. f(z)=(3z-2)/[(z-1)^2(z+1)(z-4)] has a pole of order 3 at z=1 and simple poles at z=-1 and 4
branch points.f(z) = (z − 3)^1/2 has a algebraic branch point at z = 3 and f(z) = ln(z^2 + z − 2)has logarithmic branch points where z^2 + z − 2 = 0
Essential singularity. the singularity is not pole or branch point.
singularities at infinity

Taylor’s theorem

If a=0, the series can called Maclaurin series.

Week3

for the series have negative powers of (z-z0), equations can be writen like

Laurent’s theorem

Let function f(z) be analytical in annulus R<|z-z0|<ρ, it can be writen as

Where R belongs to negative part and ρ belongs to positive part.

really conplex content, remember to work the tutorial out.

week 4

three most common form to write a complex function:
ax+by=c,
where a,b are real numbers and x,y are variables
|z-1|=|z-b|
where a,b are complex numbers
z=at+b
where t is real and a,b are complex quantities

euqation can be
(x-x0)^2+(y-y0)^2=R^2,
|z-z0|=R, z0=x0+iy0

z=x+iy=(x0+Rcosφ)+i(y0+Rsinφ)=z0+Re^iφ (0<=φ<2Pi)

integration and Cauchy’s theorem

Newton-Leibnitz formula

the equation C: z=z(t), a<=t<=b,

example:

using linear integration

if f(z)=u+iv=2+i, z=x+iy, than x=2t, y=t.

Week5

Cauchy’s theorem

for two points x1=a1+ia2, x2=b1+ib2,
z=(1-t)(a1+ia2)+t(b1+ib2) (0<t<1)
or for circuit center z0=z1+iz2, radius=R,
z=z1+iz2+R(e^iφ) (0<φ<2Pi)

to Taylor series:

Week6

Residue Theorem

Let z=z0 be a singularity of the function f(z), the coefficient a is called residue of the function f(z)

the single residue (with single power) can be calculated as

but in multiple power, the function can be changed as

for closed contour C be the boundary of domain D,

for calculating real integrals, the contour C can be drawn as

which have ignored all results with negative imaginary.

where first interial is total contour results, second one is the real integrals and third one is the curve covered this domain(most cases equals to 0)

Week7

vector calculus

scalar field: f=x+y+z
vector field: u=(p,q,r)

for scalar field f, its gradient can be written as

the functions of div and curl operators are

grad(f)=∇(f)
div(f)=∇.f
curl(f)=∇*f

initialize type	operator	result type
scalar	gradient	vector
vector	div	scalar
vector	curl	vector

like the function curl(div(u)) can not be defined cause div(u) is a scalar which can be curl which.

and there have two conditions the result will be 0:

for any scalar f, curl(grad(f))=0
for any vector u, div(curl(u))=0

for function div(grad(f))=∇.(∇(f))=fxx+fyy+fzz=∇^2(f), or called Laplancian (scalar field)
and curl(grad(f))=curl(fx,fy)=fyx-fxy=0

Week 8

a vector field u will have a potential function p (which is ∇(p)=u), only when u is conservative:
∇*u=0

if E=(2x+xy,2y+xz,2z+xy)

Week 9

Integration along curve

a curve r=(x(t),y(t),z(t)) in the range of t(a<=t<=b),

where F=∇(p)

if F is a conservative vector fild, the potential function p for F is

value if a,b,c is not stable and can be choicen what ever we want. it is only valid for conservative field