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| Front-side | Reverse-side |
|---|---|
| Cauchy-Riemann and analytic nature | It is necessary that the Cauchy/Riemann equations are satisfied for f(z) to be analytic in R If the second partial derivatives of f exist and are continuous, these are sufficient as well. |
| Cauchy-Riemann equations | (del u)/(del x) = (del v)/(del y) (del u)/(del y) = -(del v)/(del x) |
| Complex potential | SInce velocity potential phi is harmonic, there must exist a conjugate harmonic function psi(x,y) such that Omega(z) = Phi(x,y) + i*Psi(x,y) |
| Complex velocity | V = Vx + i*Vy = _(Omega’(z)) |
| Conjugate coordinates | x=(1/2)(z+z); y=(1/2i)(z-z) |
| Cross product | z1 X z2 = x1y2 – x2y1 = Im(_z1*z2) |
| Definition of differentiation | f’(z) = lim(dz->0) (f(z+dz)-f(z))/dz |
| DeMoivre’s Theorem | zn = rn(cos ntheta + i sin n*theta) |
| Dirichlet Problem | Let R be a simply-connected region bounded by asimple closed curve C. Find a function phi which satisfies Laplace’s equation and takes prescribed values on the boundary C. |
| Dot product | z1 dot z2 = x1×2 + y1y2 = Re(_z1*z2) |
| Electric field intensity | E = -grad(Phi) |
| Equipotential line | Phi(x,y) = alpha |
| Euler’s Equation | ez = e(x+ iy) = e^x(cos theta + i sin theta) |
| Exponential function | w = ez = e(x+iy) |
| Harmonic Functions | 2nd partial of u/v exists wrt x/y and are continuous, then (del2 u)(del x2) + (del2 u)(del y2) = 0 (del2 v)(del x2) + (del2 v)(del y2) = 0 |
| Heat flux | Q = -K((del Phi)/(del x) + i/(del y) = Qx + iQy |
| Isolated singular point | z = z0 is isolated if we can find delta>0 such that |z-z0| = delta encloses no singular point but z_0 |
| L’Hospital’s Rule | If f(z) and g(z) are analytic containing z0, f(z0)=g(z0)=0, but g’(z0) != 0 then lim(z->z0)f(z)/g(z) = f’(z0)/g’(z_0) |
| Laplace’s Equation | Del2 phi = (del2 phi)/(del x2) + (del2 phi)/(del y^2) |
| Linear function | w = az+b |
| Log function | w = ln z = ln r + i(theta + 2k*pi), k = 0, +-1,... |
| Neumann Problem | Let R be a simply-connected region bounded by a simple closed curve C. Find a function phi which satisfies Laplace’s equation and whose normal derivative (del phi)/(del n) takes prescribed values on the boundary C |
| Parametric equation for a line | (x – x1) = t; (y-y1) – t(y2-y1) |
| Pole | n exists such that lim(z->z0)(z-z0)^n*f(z) = a != 0 z=z_0 is a pole of order n |
| Polynomial | w = a0zn + a1z(n-1) + a2*z^(n-2) + ... + an |
| Rational algebraic function | w=P(z)/Q(z) where P and Q are polynomials |
| Roots: if w^n = z, then w= | w = r^(1/n) { cos((theta + 2kpi)/n) + i sin((theta + 2kpi)/n) } |
| Streamline | Phi(x,y) = Beta |
| Taylor series about point a | f(z) = f(a) + f’(a)(z-a) + (f’‘(a)/2!)(z-a)2 + ... + (f’n(a)/n!)(z-a)n + ... |
| Taylor series about pt a where a=0 is called | MacLaurin series |
| Velocity Potential | If Vx and Vy denote components of velocity of a fluid, there exists a function phi called the velocity potential such that Vx = (del phi)/(del x), Vy = (del phi)/(del y) |