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67 Cards in this Set

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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
8 properties
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+Δz)-f(z0)]/Δz
Δz-->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
Implied in analytic fnctns
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+2kπ

e^z=1 iff z=i2kπ
e^z1=e^z2 iff z1=z2+i2kπ
Sinz and Cosz
sinz:=(e^(iz)-e^-(iz))/(i2)
cosz:=(e^(iz)+e^-(iz))/(2)
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+i2kπ

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, Γ
single point, z0, or a finite sequence of directed smooth curves (γ1,γ2,...,γn) such that the terminal point of γk coincides with the initial point of γ(k+1) for each k=1,2,...,n-1

Γ=γ1+γ2+γ3+...+γn
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
Γ=γ1+γ2+...+γn

int over Γ of f(z)dz=int over γ1+int over γ2+int over γ3+...+int over γn
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 Γ lying in D, with initial point zI and terminal pt zT, the int of f(z) over Γ=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 Γ0 is continuously deformable to Γ1 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 Γ0
3. The function z(1,t) parameterizes the loop Γ1
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
Γ0 is continuously deformable to Γ1 in D then int over Γ0 of f(z)dz=int over Γ1 of f(z)dz
Cauchy's Integral Theorem
IF f is analytic in a simply connected domain D and Γ is any loop in D, then the int over Γ 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
Γ is a simple closed positively oriented contour, fis analytic on some simply connected domain D containing Γ and z0 is a pt inside Γ,

f(z0)-(1/i2π)int over Γ(f(z)/(z-z0)dz)
More General Version of Cauchy's Int Formula
g cont on Γ, for ea z not on Γ set

G(z):=int over Γ of [g(ζ)/(ζ-z)dζ]

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

d^nG(z)/dz^n=(n!/i2π)(int over Γ of [g(ζ)/(ζ-z)^(n+1)dζ]
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