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