AMath 501

Outline

Basic Vector Manipulations

Coordinates & Position Vectors

  1. Cartesian Coordinates:
    • r=xi^+yj^+zk^
  2. Spherical Coordinates:
    • r=re^r
  3. Cylindrical Coordinates:
    • r=re^r+ze^z

Curvilinear Coordinates

e^i=r/uir/ui,i=1,2,3 hi=rui

Derivatives and Arc Length

  1. Derivative of position vector dr:
dr=ru1du1+ru2du2+ru3du3 dr=h1du1e^1+h2du2e^2+h3du3e^3
  1. Arc length differential (ds)2:
(ds)2=drdr=h12(du1)2+h22(du2)2+h32(du3)2
  1. Arc Length Formula:
L=ab|dr(t)dt|dt

Gradient Operator

:=i^x+j^y+k^z(in Cartesian):=1h1e^1u1+1h2e^2u2+1h3e^3u3

Divergence Operator

  1. In Cartesian coordinates:
divF=F=F1x+F2y+F3z
  1. In general:
divF=1h1h2h3[u1(F1h2h3)+u2(F2h1h3)+u3(F3h1h2)]

where:

F=F1e^1+F2e^2+F3e^3
  1. Write down div in spherical/cylindrical coordinates.

Curl Operator

CurlF=1h1h2h3|h1e^1h2e^2h3e^3u1u2u3h1F1h2F2h3F3|

Physical Explanation

Line Integral

W=r(a)r(b)Fdr=r(a)r(b)[F1(h1du1)+F2(h2du2)+F3(h3du3)] W=abFdrdtdt=ab[F1h1du1dt+F2h2du2dt+F3h3du3dt]dt

Conservative Field

W=r(a)r(b)Fdr=dΦ=Φ(r(b))Φ(r(a))

Surface Integral

SFn^dson a surfaceS dS=n^ds=(ru1×ru2)du1du2=ru1×ru2ru1×ru2ru1×ru2du1du2=n^ds n^=ff,which is easier than:n^=rx×ryrx×ry

Surface Representation

ds=1+(hx)2+(hy)2dxdy r=xi^+yj^+h(x,y)k^

and:

rx=i^+hxk^,ry=j^+hyk^n^ds=f(fx)2+(fy)2+(fz)21+(hx)2+(hy)2dxdy n^ds=fdxdy

Surface Area

A=ds

Divergence Theorem

VFn^ds=VdivFdV dV=h1h2h3du1du2du3

Stokes' Formula/Theorem

ScurlFn^ds=CFdr,

This expands to:

×F=(F3yF2z)i^(F3xF1z)j^+(F2xF1y)k^

In 2D:

curlF=F2xF1y

Definition of Holomorphic Functions / Analytic Functions

A holomorphic function (or analytic function) is a complex function f(z)  that is differentiable at every point in an open subset of the complex plane C . Differentiability in the complex sense means that the limit defining the derivative:

f(z)=limh0f(z+h)f(z)h

exists and is the same no matter how h approaches 0 in the complex plane.

Cauchy-Riemann Relations:

For f(z)=u(x,y)+iv(x,y)  (where z=x+iy), the function is holomorphic if:

ux=vy,uy=vx.

Branch Cuts / Points:

Some functions (like lnz or z) are not holomorphic everywhere because they involve multivalued branches. For example:

Example

  1. Holomorphic Function: f(z)=z2+i

Here, u(x,y)=x2y2  and v(x,y)=2xy+1. These satisfy the Cauchy-Riemann equations:

ux=2x,vy=2x

and

uy=2y,vx=2y.
  1. Non-Holomorphic Function: f(z)=|z|2=x2+y2

Here, u(x,y)=x2+y2 and v(x,y)=0. The Cauchy-Riemann equations fail:

ux=2xvy=0.

Cauchy Theorem

If f(z) is holomorphic (analytic) in a simply connected domain D and γ is a closed contour in D, then:

γf(z)dz=0.

Key Idea:
The integral of an analytic function over a closed loop is zero if the function is analytic everywhere inside and on the contour.

Example:

Let f(z)=z2, and γ is the circle |z|=1:

γz2dz=0

since f(z) is holomorphic everywhere.

Cauchy Integral Formula

If f(z) is holomorphic in a simply connected domain D and γ is a simple closed contour in D, then for any point a inside γ:

f(a)=12πiγf(z)zadz.

Key Idea:
The value of a holomorphic function inside a contour can be computed using the values of the function on the contour.

Example:

Let f(z)=z2, and γ is the circle |z|=2. Compute f(0):

f(0)=12πiγz2zdz=12πiγzdz.

Since γzdz=0, f(0)=0.

Residue

Intuition: Circulation Around the Singularity

The residue measures how much the function “circulates” around the singularity. If you imagine walking along the contour C, the residue tells you the strength and direction of this “circulation” (positive for counterclockwise, negative for clockwise).

In f(z)=1z1, the residue is 1, meaning a unit of circulation counterclockwise around z=1.

Key Points About Residues and Singularities

  1. Residue Explains Local Behavior Near a Singularity
    The residue at a singularity z=z0 describes the “circulation” of f(z) near that singularity.
    For example, in f(z)=1(z+2)(z1):
  1. Residues Are Independent of Distance From the Singularity
    Residues depend only on the singularity itself, not on the distance from it. The contour C used to calculate the residue can have any radius, as long as it encloses only that singularity (and no others).

This is because the integral Cf(z)dz over any closed contour enclosing the singularity gives the same result, thanks to the Cauchy Integral Formula.

  1. Residue Measures Local Circulation
    Residue tells us how f(z) “circulates” locally around the singularity. This is like isolating the behavior of f(z) at that point.
  1. Why Distance Doesn’t Affect Residue
    The residue at a singularity depends only on the coefficient of 1zz0 in the Laurent series expansion. Since Laurent series expansions focus on the singularity and ignore the behavior elsewhere, the residue is unaffected by the distance from the singularity.

Residue Theorem

Let f(z) be holomorphic in a domain D, except for isolated singularities. If γ is a simple closed contour in D that encloses these singularities, then:

γf(z)dz=2πiResidues of f(z) inside γ.

Key Idea:
The integral around a closed contour is determined by the sum of residues of the function’s singularities inside the contour.

Single Singular Point at  z = 0

In your specific example, the contour C is the unit circle |z|=1, and the function f(z) has only one singularity inside the contour, at z=0. For this special case:

Cf(z)dz=2πiRes(f(z),0).

The residue of