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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*=aib



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=(zz*)/i2 6. (z*)*=z 7. z=z* 8. zz*=z² 
8 properties


Triangle Inequality

z1+z2<=z1+z2
>z2z1<=z1z2 


Polar Form

z=x+iy=r(cosθ+isinθ)
r=z=(x²+y²)^.5 θ=argz=tan^(1)(y/x) 


Properties of Polar Form

z1z2=z1z2=r1r2
argz1z2=argz1+argz2=θ1+θ2 argz*=argz example: z1/z2=(r1/r2)cis(θ1θ2) 


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/(z²+1) x2=2Imz/(z²+1) x3=(z²1)/(z²+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
CR 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 CR 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' 


CauchyRiemann 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(zζ1)^d1(zζ2)^d2...(zζn)^dn r(z)=A0^1/((zζ1)^d1)+A1^1/((zζ1)^(d11))+...+A(d11)^1/(zζ1)+A0^2/(zζ2)^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^ze^z)/(i2)
coshz:=(e^z+e^z)/2 


Logarithmic Function

logz:=Logz+iargz
=Logz=iArgz+i2kπ Logz:=Logz=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,...,n1
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 complexcalued 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)/(zz0)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)/(ζz)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)/(ζz)^(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=0inf c^j converges to 1/(1c) if c<1
1/(1c)(1+c+c²+c³+...+c^(n1)+c^n)=c^(n+1)/(1c) 


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!*(zz0)^n



Taylor Series of an Analytic Fnctn

if f is analytic in the disk zz0<R, then the taylor series converges to f(z) for all z in the disk and the convergence is uniform in any closed subdisk zz0<=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(zz0)^j aj=d^jf(z0)/dz^j/j!
g(z)=Σbj(zz0)^j bj=d^jg(z0)/dz^j/j! the T.S. for cf(z)=Σcaj(zz0)^j and the T.S. for f(z)±g(z)=Σ(aj±bj)(zz0)^j 


Cauchy Product

f(z)=Σaj(zz0)^j aj=d^jf(z0)/dz^j/j!
g(z)=Σbj(zz0)^j bj=d^jg(z0)/dz^j/j! f(z)g(z)=Σcj(zz0)^j cj=Σa(jL)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^(jL)f/dz^(jL)/(jL)!*d^L(g)/dz^L/L!



Power Series

Σaj(zz0)^j



Convergence of Power Series

for any power series Σaj(zz0)^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 zz0<R 2. the series converges uniformly in any closed subdisk zz0<=R'<R 3. the series diverges for zz0>R The nuumber R is called the radius of convergence of the power series 
