Math 282 Final

Continuity of power series

A power series represents a continuous function S(z) at each point inside its circle of convergence |z − z0| = R.

maximum modulus principle

If a function f is analytic and not constant in a given domain D, then |f (z)| has no maximum value in D. That is, there is no point z0 in the domain such that | f (z)|≤| f (z0)| for all points z in it. Suppose that a function f is continuous on a closed bounded region R and that it is analytic and not constant in the interior of R. Then the maximum value of | f (z)| in R, which is always reached, occurs somewhere on the boundary of R and never in the interior.

Cauchy-Goursat theorem.

If a function f is analytic at all points interior to and on a simple closed contour C, then ∫f=0 also If a function f is analytic throughout a simply connected domain D, then ∫f=0 for all C in D

residue at infinity

If a function f is analytic everywhere in the finite plane except for a finite number of singular points interior to a positively oriented simple closed contour C, then ∫f=2πiResz=0[1/z^2f(1/z)]

Louvilles theorem

If a function f is entire and bounded in the complex plane, then f (z) is constant throughout the plane.

absolute and uniform convergence

If a power series converges when z=z1 (z1≠z0) then it is absolutely convergent in open disk |z-z0|<|z1-z0| If z1 is a point inside the circle of convergence |z − z0| = R of a power series then that series must be uniformly convergent in the closed disk |z − z0| ≤ |z1 − z0|

accumulation point

if each deleted neighborhood of z0 contains at least one point of S.

meromorphic

if it is analytic throughout D except for poles.

integrals

integrals of the components u and v

Cauchy principle value

lim R→∞∫R to -R f(x)dx

limit

lim z→z0 f(z) = w0

Continuous at a point

lim z→z0 f(z) exists f (z0) exists lim z→z0 f(z) = f (z0).

Limit of components

lim(x,y)→(x0,y0) u(x, y) = u0 and lim (x,y)→(x0,y0) (1) v(x, y) = v0 then lim(x,y)→(x0,y0) f(x,y)=u0+iv0

Logarithm

log z = ln |z| + i arg z log z = ln r + i(θ + 2nπ)

Let z0 be an isolated singular point of a function f . The following two statements are equivalent:

z0 is a pole of order m (m = 1, 2, . . .) of f (z) can be written in the form f (z) = φ(z)/(z − z0)^m (m = 1, 2, . . .), where φ(z) is analytic and nonzero at z0. Moreover, if statements (a) and (b) are true, Res z=z0 f (z) = φ(z0) when m = 1 Res z=z0 f (z) = φ^(m−1) (z0)/(m − 1)! when m = 2, 3,...

polar form

z=r(cosθ+isinθ) r=|z| θ=arctan(y/x)= arg(z)

exponential form

z=z₀+re^iθ

power function z^c

z^c = e^clog z

Triangle Inequality

|z1 + z2|≤|z1|+|z2| also, |z1 + z2| ≥ ||z1|−|z2||.

z*z

|z|^2

|z|

√(x^2+y^2)

Residue from Laurent

∫f (z) dz = 2πi Res z=z0f (z)=bn. bn = 1/2πi ∫ f (z) dz (z − z0)^(−n+1)

counter integral

∫ₙf(z)=∫f[z(t)]z'(t) straight line: z(t) = (1 − t)z1+tz2 with 0 ≤ t ≤ 1. arc: z(θ) = z0 + R e iθ with α ≤ θ ≤ β.

z*

x-iy

Harmonic

Hxx (x, y) + Hyy (x, y) = 0, if f is analytic u and v are harmonic

Let f denote a function that is analytic at a point z0. The following two statements are equivalent:

(a) f has a zero of order m at z0; (b) there is a function g, which is analytic and nonzero at z0 , such that f (z) = (z − z0)^m g(z).

z^-1

(x/√(x^2+y^2),-y/√(x^2+y^2)

Arg(z)

-π<θ≤π

deleted neighborhood

0 < |z − z0| < ε

analytic at a point

0 if it is analytic in some neighborhood of z0.

winding number

1/2π ∆C arg f (z)

roots of z₀

= r₀^(1/n)exp^i(θ₀/n + 2kπ/n) k=0,...,n-1 r^n =r0, nθ = θ0 + 2kπ

multiplication and division of power series

f (z)g(z) = a0b0 + (a0b1 + a1b0)(z − z0) + (a0b2 + a1b1 + a2b0)(z − z0)^2 f (z)g(z)] (n) = ∑(n k)f^(k)(z)g^(n−k)(z) (n k) = n!/(k!(n-k!))

Derivative

f'(z0)= lim z→z0 (f(z) − f (z0))/(z − z0)

arg(z)

Arg(z)+2nπ

Bounded set

Every point in set S lies in come circle |z|=R

Interior, exterior, boundary points

Interior: when 0 < |z − z0| < ε lies entirely inside a set S Exterior: when 0 < |z − z0| < ε lies entirely outside a set S Boundary: Not interior or exterior point

Cauchy's residue theorem

Let C be a simple closed contour, described in the positive sense. If a function f is analytic inside and on C except for a finite number of singular points zk (k = 1, 2,..., n) inside C then ∫C f(z)=2πi∑ Res z=zk f (z)

integration of power series

Let C denote any contour interior to the circle of convergence of the power series (1), and let g(z) be any function that is continuous on C. The series formed by multiplying each term of the power series by g(z) can be integrated term by term over C; that is, ∫g(z)S(z)dz=∑aₙ∫g(z)(z − z0)^n

cauchy integral formula

Let f be analytic everywhere inside and on a simple closed contour C, taken in the positive sense. If z0 is any point interior to C, then f^n(z0) = n!/2πi∫f (z)/(z − z0)^(n+1)

3 types of singularities

Removable Singular Points: When every bn is zero, so that you have normal taylor series (Note that the residue at a removable singular point is always zero.) Essential Singular Points: If an infinite number of the coefficients bn in the principal part (2) are nonzero, z0 is said to be an essential singular point of f. Poles of Order m: If the principal part of f at z0 contains at least one nonzero term but the number of such terms is only finite, then there exists a positive integer m (m ≥ 1) such that bm = 0 and bm+1 = bm+2 =···= 0. That is, expansion (1) takes the form

analytic in open set

S if it has a derivative everywhere in that set.

Cuachy's inequality

Suppose that a function f is analytic inside and on a positively oriented circle CR, centered at z0 and with radius R (Fig. 71). If MR denotes the maximum value of |f(z)| on CR, then |f^(n)(z0)| ≤ n!MR/R^n

Taylor Series

Suppose that a function f is analytic throughout a disk |z − z0| < R0, centered at z0 and with radius R0 Then f (z) has the power series representation f (z) = ∑an(z − z0)^n an=f&n(z0)/n!

Laurents Theorem

Suppose that a function f is analytic throughout an annular domain R1 < |z − z0| < R2 , centered at z0 , and let C denote any positively oriented simple closed contour around z0 and lying in that domain. Then, at each point in the domain, f (z) has the series representation f (z) = ∑cn(z − z0)^n from -inf to inf for (R1 < |z − z0| < R2) cn = 1/2πi ∫f (z)/(z − z0)^n+1

Riemann's theorem

Suppose that a function f is bounded and analytic in some deleted neighborhood 0 < |z − z| < ε of z0. If f is not analytic at z0, then it has a removable singularity there.

Casorati-Weierstrass theorem

Suppose that z0 is an essential singularity of a function f , and let w0 be any complex number. Then, for any positive number ε, the inequality (3) | f (z) − w0| < ε is satisfied at some point z in each deleted neighborhood 0 < |z − z0| < δ of z0

entire

entire function is a function that is analytic at each point in the entire plane.

Rouche's theorem

Theorem. Let C denote a simple closed contour, and suppose that (a) two functions f (z) and g(z) are analytic inside and on C; (b) | f (z)| > |g(z)| at each point on C. Then f (z) and f (z) + g(z) have the same number of zeros, counting multiplicities, inside C.

Jordan's lemma

Theorem. Suppose that (a) a function f (z) is analytic at all points in the upper half plane y ≥ 0 that are exterior to a circle |z| = R0; (b) CR denotes a semicircle z = Reiθ (0 ≤ θ ≤ π), where R > R0 (c) for all points z on CR, there is a positive constant MR such that | f (z)| ≤ MR and lim R→∞ MR = 0. Then, for every positive constant a, lim R→∞∫ CR f (z)e^iaz dz = 0.

Cuachy Reimann Equations

Ux=Vy, Uy=-Vx, rUr=Vθ, Uθ=-rVr,

e^z

ez = e^(x)e^(iy)

Connected set S

all points can be connected with a line that lies entirely in S

simple arc, simple closed curve, positively oriented

arc that doesnt cross itself, arc that doesnt cross itself with connecting endpoints counterclockwise

arg(z1z2)

arg(z1)+arg(z2)=(θ1 + θ2) + 2nπ also argz1/z2 = arg z1 − arg z2

Closed set

contains all boundary points

e^iθ

cosθ+isinθ

trig functions

sin x = e^iz−e^−iz/2i cos x = e^iz+e^−iz/2 . sinh x = e^z−e^−z/2 cosh x = e^z+e^−z/2 . |sin z|^2 = sin^2 x + sinh^2 y, | cosz|^2 = cos^2 x + sinh^2y.