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67 Cards in this Set
- Front
- Back
- 3rd side (hint)
Complex Number
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a+ib
a,b real numbers |
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Modulus: |z|
(z=a+ib) |
(a²+b²)^.5
a imaginary part b real part |
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Complex Conjugate
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z*=a-ib
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Properties of Complex Conjugate
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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
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Triangle Inequality
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|z1+z2|<=|z1|+|z2|
-->|z2|-|z1|<=|z1-z2| |
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Polar Form
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z=x+iy=r(cosθ+isinθ)
r=|z|=(x²+y²)^.5 θ=argz=tan^(-1)(y/x) |
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Properties of Polar Form
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|z1z2|=|z1||z2|=r1r2
argz1z2=argz1+argz2=θ1+θ2 argz*=-argz example: z1/z2=(r1/r2)cis(θ1-θ2) |
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Complex Exponential
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z=x+iy
e^(z)=e^x(cosy+isiny) |
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De Moivre's formula
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(cosθ+isinθ)^n=cosnθ+isinnθ n=1,2,3,...
because: (e^iθ)^n=e^inθ |
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Domain
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Open, connected set
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u(x,y)=constant in D
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δu/δx=δu/δy=0
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Stereographic Projection and Riemann Sphere
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x1²+x2²+x3²=1
x1=2Rez/(|z|²+1) x2=2Imz/(|z|²+1) x3=(|z|²-1)/(|z|²+1) |
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Limits
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lim zn=z0
n-->inf |
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Continuity
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f is continuous at z0 if
lim f(z)=f(z0) z-->z0 |
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Properties of Limits
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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 |
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Properties of Continuity
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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
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Analytic
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every point on an open set has a derivative
C-R eqns must hold at every pt of open set |
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Derivative
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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. |
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Properties of Derivatives
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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' |
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Cauchy-Riemann Eqns
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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
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Constant Function
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if f is analytic, f'=0 in domain D
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Harmonic Functions
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if in D, all 2nd order partials of f are continuous and at each pt in D, f satisfies the 2d laplace eqn
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2 Dimensional Laplace Eqn
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δ²f/δx²+δ²f/δy²=0
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Polynomials and Rational Functions
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p(z)=a0+a1z+a2z²+a3z³+...amz^m
q(z)=b0+b1z+b2z²+b3z³+...bnz^n r(z)=p(z)/q(z) |
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Entire
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analytic on the entire complex plane
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Fundamental Theorem of Algebra
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every nonconstant polynomial with complex coefficients has at least one zero in C
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Partial Fraction Decomposition
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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+... |
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Exponential Function
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|e^z|=e^x
arg e^z=y+2kπ e^z=1 iff z=i2kπ e^z1=e^z2 iff z1=z2+i2kπ |
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Sinz and Cosz
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sinz:=(e^(iz)-e^-(iz))/(i2)
cosz:=(e^(iz)+e^-(iz))/(2) |
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Sinz and Cosz
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sinz:=(e^(iz)-e^-(iz))/(i2)
cosz:=(e^(iz)+e^-(iz))/(2) |
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Sinhz and Coshz
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sinhz:=(e^z-e^-z)/(i2)
coshz:=(e^z+e^-z)/2 |
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Logarithmic Function
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logz:=Log|z|+iargz
=Log|z|=iArgz+i2kπ Logz:=Log|z|=iArgz |
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Complex Powers
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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 |
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Smooth Arc
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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 |
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Contour, Γ
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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 |
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Jordan Curve Theorem
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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
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Contour Integral Theorem
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If f is continuous on the directed smooth curve γ, then if f is intergrable along γ.
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Generalized form of the Fundamental Theorem of Calculus
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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)
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Integral using Parameterization
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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 |
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Summing of Contours
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Γ=γ1+γ2+...+γn
int over Γ of f(z)dz=int over γ1+int over γ2+int over γ3+...+int over γn |
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Independence of Path
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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)
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Properties of Continuous Functions
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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 |
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Continuously Deformable
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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 |
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Simply Connected Domain
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Any D possesing the property that every loop in D can be continuously deformed in D to a pt
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Deformation Invariance Theorem
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Γ0 is continuously deformable to Γ1 in D then int over Γ0 of f(z)dz=int over Γ1 of f(z)dz
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Cauchy's Integral Theorem
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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
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Properties of Analytic Fnctns Concluded Using Cauchy's Theorem
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An analytic fnctn in a simply connected domain....
-has an antiderivative -its contour integrals are independent of path -loop integrals=0 |
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Cauchy's Integral Formula
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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) |
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More General Version of Cauchy's Int Formula
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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] |
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Derivatives of Analytic Fnctns
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if f is analytic on D, then all of its derivatives, f',f'',f''',...,d^nf/dz^n exist and are analytic on D
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Liouville Theorem
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the only bounded entire fnctns are the constant fnctns
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Fundamental Theorem of Algebra II
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every nonconstant polynomial with the complex coefficients has at least one zero
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Maximum Modulus Principle
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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
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Max Modulus Principle Part II
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a functn analytic in a bounded domain and cont up to and including its boundary attains its max modulus on the boundary
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Convergence of a Series
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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) |
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Comparison Test
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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
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Ratio Test
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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
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Uniform Convergence
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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 |
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Telescoping Series
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Σ[1/(n+2)-1/(j+1)]
conv if lim = 0 n-->inf |
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Taylor Series
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Σ(d^nf/dz^n)(z0)/n!*(z-z0)^n
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Taylor Series of an Analytic Fnctn
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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
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Derivative of Taylor Series
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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.
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Properties of Taylor Series
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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 |
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Cauchy Product
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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 |
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Leibniz's Formula
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d^j(fg)/dz^j=Σj!d^(j-L)f/dz^(j-L)/(j-L)!*d^L(g)/dz^L/L!
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Power Series
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Σaj(z-z0)^j
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Convergence of Power Series
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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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