Functions of a Complex Variable

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Complex Number: a+ib

a,b real numbers
Modulus: |z| (z=a+ib): (a²+b²)^.5

a imaginary part
b real part
Complex Conjugate: z*=a-ib
Properties of Complex Conjugate: a real number
z1,z2 complex numbers

1. a*=a
2. (z1z2)*=z1*z2*
3. (z1/z2)*=z1*/z2*, z2=/0
4. Rez=(z+z*)/2
5. Imz=(z-z*)/i2
6. (z*)*=z
7. |z|=|z*|
8. zz*=|z|²
Triangle Inequality: |z1+z2|<=|z1|+|z2|
-->|z2|-|z1|<=|z1-z2|
Polar Form: z=x+iy=r(cosθ+isinθ)
r=|z|=(x²+y²)^.5
θ=argz=tan^(-1)(y/x)
Properties of Polar Form: |z1z2|=|z1||z2|=r1r2
argz1z2=argz1+argz2=θ1+θ2
argz*=-argz

example:
z1/z2=(r1/r2)cis(θ1-θ2)
Complex Exponential: z=x+iy
e^(z)=e^x(cosy+isiny)
De Moivre's formula: (cosθ+isinθ)^n=cosnθ+isinnθ n=1,2,3,...
because:
(e^iθ)^n=e^inθ
Domain: Open, connected set
u(x,y)=constant in D: δu/δx=δu/δy=0
Stereographic Projection and Riemann Sphere: x1²+x2²+x3²=1

x1=2Rez/(|z|²+1)
x2=2Imz/(|z|²+1)
x3=(|z|²-1)/(|z|²+1)
Limits: lim zn=z0
n-->inf
Continuity: f is continuous at z0 if

lim f(z)=f(z0)
z-->z0
Properties of Limits: lim f(z)=A lim g(z)=B
z-->z0 z-->z0
1. lim (f(z)±g(z))=A±B
z-->z0
2. lim f(z)g(z)=AB
z-->z0
3. limf(z)/g(z)=A/B B=/0
z-->z0
Properties of Continuity: If f(z) and g(z) are continuous at z0, then so are f(z)±g(z), f(z)g(z), and f(z)/g(z) given g(z0)=/0
Analytic: every point on an open set has a derivative

C-R eqns must hold at every pt of open set
Derivative: df/dz(z0)=
f'(z0):=lim [f(z0+Î”z)-f(z0)]/Î”z
Î”z-->0

for a fnctn to be differentiable at a pt z0, the C-R eqns must apply at z0.
Properties of Derivatives: 1. (f±g)'(z)=f'(z)±g'(z)
2. (cf)'(z)=cf'(z)
3. (fg)'=fg'+f'g
4. (f/g)'=(gf'-fg')/g²
5. d/dz(f(g))=f'(g)g'
Cauchy-Riemann Eqns: δu/δx=δv/δy
δu/δy=-δv/δx

Are not enough to ensure differentiability... needs to make sure 1st partials of u and v are contiuous at z0
Constant Function: if f is analytic, f'=0 in domain D
Harmonic Functions: if in D, all 2nd order partials of f are continuous and at each pt in D, f satisfies the 2d laplace eqn
2 Dimensional Laplace Eqn: δ²f/δx²+δ²f/δy²=0
Polynomials and Rational Functions: p(z)=a0+a1z+a2z²+a3z³+...amz^m
q(z)=b0+b1z+b2z²+b3z³+...bnz^n
r(z)=p(z)/q(z)
Entire: analytic on the entire complex plane
Fundamental Theorem of Algebra: every nonconstant polynomial with complex coefficients has at least one zero in C
Partial Fraction Decomposition: r(z)=(a0+a1z+a2z²+a3z³+...+amz^m)/
bn(z-ζ1)^d1(z-ζ2)^d2...(z-ζn)^dn

r(z)=A0^1/((z-ζ1)^d1)+A1^1/((z-ζ1)^(d1-1))+...+A(d1-1)^1/(z-ζ1)+A0^2/(z-ζ2)^d2+...
Exponential Function: |e^z|=e^x
arg e^z=y+2kϬ

e^z=1 iff z=i2kϬ
e^z1=e^z2 iff z1=z2+i2kϬ
Sinz and Cosz: sinz:=(e^(iz)-e^-(iz))/(i2)
cosz:=(e^(iz)+e^-(iz))/(2)
Sinhz and Coshz: sinhz:=(e^z-e^-z)/(i2)
coshz:=(e^z+e^-z)/2
Logarithmic Function: logz:=Log|z|+iargz
=Log|z|=iArgz+i2kϬ

Logz:=Log|z|=iArgz
Complex Powers: z^α=(e^logz)^α=e^(αlogz)

α is real integer--> single value
α is real, rational--> finite #of values
α is anything else--> inf # of values
Smooth Arc: z=z(t), a<=t<=b

1. z(t) has continuous derivative on
[a,b]
2. z'(t) never vanishes on [a,b]
3. z(t) is 1:1 on [a,b]

3'. z(t0 is 1:1 on the half open
interval [a,b), but z(b)=z(a) and
z'(b)=z'(a) if SMOOTH CLOSED CURVE
Contour, Î“: single point, z0, or a finite sequence of directed smooth curves (Î³1,Î³2,...,Î³n) such that the terminal point of Î³k coincides with the initial point of Î³(k+1) for each k=1,2,...,n-1

Î“=Î³1+Î³2+Î³3+...+Î³n
Jordan Curve Theorem: Any simple closed contour separates the plane into 2 domains, each having the curve as its boundary. The interior domain is bounded and the exterior domain is unbounded
Contour Integral Theorem: If f is continuous on the directed smooth curve γ, then if f is intergrable along γ.
Generalized form of the Fundamental Theorem of Calculus: If the complex-calued function f is cont on [a,b] and F'(t)=f(t) for all t on [a,b], then integral over [a,b] of f(t)dt=F(b)-F(a)
Integral using Parameterization: if f is a cont fnctn on directed smooth curve γ for z=z(t) for a<=t<=b

int over γ of f(z)dz= int over [a,b] of f(z(t))z'(t)dt
Summing of Contours: Î“=Î³1+Î³2+...+Î³n

int over Î“ of f(z)dz=int over Î³1+int over Î³2+int over Î³3+...+int over Î³n
Independence of Path: Suppose that the f(z) is continuous in a domain D and has a antiderivative F(z)throughout D. THen for any contour in Î“ lying in D, with initial point zI and terminal pt zT, the int of f(z) over Î“=F(zT)-F(zI)
Properties of Continuous Functions: f is cont on D(domain)

1. f has an antiderivative in D
2. every loop integral of f in D vanishes
3. the contour integrals of f are independent of path in D
Continuously Deformable: loop Î“0 is continuously deformable to Î“1 in D if there exists a fnctn z(s,t) continuous on the unit square 0<=s<=1, 0<=t<=1, that satisfies the following conditions:

1. FOr ea fixed s in [0,1], the fnctn z(s,t) parameterizes a loop lying in D
2. The function z(0,t) parameterizes the loop Î“0
3. The function z(1,t) parameterizes the loop Î“1
Simply Connected Domain: Any D possesing the property that every loop in D can be continuously deformed in D to a pt
Deformation Invariance Theorem: Î“0 is continuously deformable to Î“1 in D then int over Î“0 of f(z)dz=int over Î“1 of f(z)dz
Cauchy's Integral Theorem: IF f is analytic in a simply connected domain D and Î“ is any loop in D, then the int over Î“ of f(z)dz=0
Properties of Analytic Fnctns Concluded Using Cauchy's Theorem: An analytic fnctn in a simply connected domain....
-has an antiderivative
-its contour integrals are independent of path
-loop integrals=0
Cauchy's Integral Formula: Î“ is a simple closed positively oriented contour, fis analytic on some simply connected domain D containing Î“ and z0 is a pt inside Î“,

f(z0)-(1/i2Ï€)int over Î“(f(z)/(z-z0)dz)
More General Version of Cauchy's Int Formula: g cont on Î“, for ea z not on Î“ set

G(z):=int over Î“ of [g(Î¶)/(Î¶-z)dÎ¶]

then G is analytic at ea z not on Î“ and its derivative is

d^nG(z)/dz^n=(n!/i2Ï€)(int over Î“ of [g(Î¶)/(Î¶-z)^(n+1)dÎ¶]
Derivatives of Analytic Fnctns: if f is analytic on D, then all of its derivatives, f',f'',f''',...,d^nf/dz^n exist and are analytic on D
Liouville Theorem: the only bounded entire fnctns are the constant fnctns
Fundamental Theorem of Algebra II: every nonconstant polynomial with the complex coefficients has at least one zero
Maximum Modulus Principle: if f is analytic in D and |f(z)| achieves its max value at a pt z0 in D, then f is constant in D
Max Modulus Principle Part II: a functn analytic in a bounded domain and cont up to and including its boundary attains its max modulus on the boundary
Convergence of a Series: Σj=0-inf c^j converges to 1/(1-c) if |c|<1

1/(1-c)-(1+c+c²+c³+...+c^(n-1)+c^n)=c^(n+1)/(1-c)
Comparison Test: If the terms cj satisfy the inequality |cj|<=Mj for all int j larger than some number J. Then if the series ΣMj converges, so does Σcj
Ratio Test: Suppose the terms of Σcj have the property that the ratios |cj+1/cj| approaches a limit L as j-->inf. The the series converges if L<1 and diverges if L>1
Uniform Convergence: The sequence {Fn(z)} is said to converge uniformly to F(z) on the set T if for any ε>0 there exists an integer N s.t. when n>N,
|F(z)-Fn(z)|<ε for all z in T
Telescoping Series: Σ[1/(n+2)-1/(j+1)]

conv if lim = 0
n-->inf
Taylor Series: Σ(d^nf/dz^n)(z0)/n!*(z-z0)^n
Taylor Series of an Analytic Fnctn: if f is analytic in the disk |z-z0|<R, then the taylor series converges to f(z) for all z in the disk and the convergence is uniform in any closed subdisk |z-z0|<=R'<R
Derivative of Taylor Series: if f is analytic at z0 (i.e. Taylor series exists), then f' can be obtained by termwise differentiation of the Taylor series for f around z0 and converges in the same disk as the series for f.
Properties of Taylor Series: f(z)=Σaj(z-z0)^j aj=d^jf(z0)/dz^j/j!
g(z)=Σbj(z-z0)^j bj=d^jg(z0)/dz^j/j!

the T.S. for cf(z)=Σcaj(z-z0)^j
and the T.S. for f(z)±g(z)=Σ(aj±bj)(z-z0)^j
Cauchy Product: f(z)=Σaj(z-z0)^j aj=d^jf(z0)/dz^j/j!
g(z)=Σbj(z-z0)^j bj=d^jg(z0)/dz^j/j!

f(z)g(z)=Σcj(z-z0)^j cj=Σa(j-L)bL L=[0,j]

Convergence: converges at least to the smaller of the two disks
Leibniz's Formula: d^j(fg)/dz^j=Σj!d^(j-L)f/dz^(j-L)/(j-L)!*d^L(g)/dz^L/L!
Power Series: Σaj(z-z0)^j
Convergence of Power Series: for any power series Σaj(z-z0)^j there is a real number R between 0 and inf, inclusive, which depends only on the coefficients {aj}, s.t.

1. the series converges for |z-z0|<R
2. the series converges uniformly in any closed subdisk |z-z0|<=R'<R
3. the series diverges for |z-z0|>R

The nuumber R is called the radius of convergence of the power series

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