Cauchy

Computing the residuew of a monomial

∮
C

(
z
−

z
0

)
n

d
# z

∫
0

2
π

e

i
n
θ

⋅
i

e

i
θ

d
# θ
i

∫
0

2
π

e

i
(
n
1
)
θ

d
# θ

{

2
π
i

if
# n
−
1

0

otherwise

{\displaystyle
\begin{align}
\oint\_C (z-z\_0)^n dz = \int\_0^{2\pi} e^{in \theta} \cdot i e^{i \theta} d \theta = i \int\_0^{2\pi} e^{i (n+1) \theta} d\theta
= \begin{cases}
2\pi i & \text{if } n = -1 \\
0 & \text{otherwise}
\end{cases}
\end{align}
}

where

C

{\displaystyle C}

 is the circle with radius

1

{\displaystyle 1}

 therefore

z
→

e

i
θ

{\displaystyle z \to e^{i\theta}}

 and

d
z
→
d
(

e

i
θ

# )
i

e

i
θ

d
θ

{\displaystyle dz \to d(e^{i\theta}) = ie^{i\theta}d\theta}

∮

C
r

f
(
z
)

z
−

z
0

d
# z

∮

C
r

f
(

z
0

)

z
−

z
0

d
z

∮

C
r

f
(
z
)
−
f
(

z
0

)

z
−

z
0

d
# z
f
(

z
0

)
2
π
i
0

{\displaystyle \oint\_{C\_r}\frac{f(z)}{z-z\_0}dz = \oint\_{C\_r}\frac{f(z\_0)}{z-z\_0}dz + \oint\_{C\_r}\frac{f(z)-f(z\_0)}{z-z\_0}dz = f(z\_0)2\pi i + 0}

The last term in the equation above equals zero when r=0. Since its value is independent of r it must therefore equal zero for all values of r.

∫
Γ

f
(
z
)
⋅
d
z

≤
M
a
x
(

f
(
z
)

)
⋅
l
e
n
g
t
h
(
Γ
)

Cauchy's integral formulaw states that the value of a holomorphic function within a disc is determined entirely by the values on the boundary of the disc.

Divergence can be nonzero outside the disc.

Cauchy's integral formula can be generalized to more than two dimensions.

f

(
0
)

(

z
0

# )

1

2
π
i

∮
γ

f
(
z
)

1

z
−

z
0

d
z

{\displaystyle
f^{(0)}(z\_0)=\dfrac{1}{2\pi i}\oint\_\gamma f(z)\frac{1}{z-z\_0}dz
}

Which gives:

f
′

(

z
0

# )

1

2
π
i

∮
γ

f
(
z
)

1

(
z
−

z
0

)
2

d
z

{\displaystyle
f'(z\_0)=\dfrac{1}{2\pi i}\oint\_\gamma f(z)\frac{1}{(z-z\_0)^2}dz
}

f
″

(

z
0

# )

2

2
π
i

∮
γ

f
(
z
)

1

(
z
−

z
0

)
3

d
z

{\displaystyle
f''(z\_0)=\dfrac{2}{2\pi i}\oint\_\gamma f(z)\frac{1}{(z-z\_0)^3}dz
}

f

(
n
)

(

z
0

# )

n
!

2
π
i

∮
γ

f
(
z
)

1

(
z
−

z
0

)

n
1

d
z

{\displaystyle f^{(n)}(z\_0) = \frac{n!}{2\pi i} \oint\_\gamma f(z)\frac{1}{(z-z\_0)^{n+1}}\, dz}

Note that n does not have to be an integer. See Fractional calculus.
The Taylor series becomes:

f
(
z

# n
0

∞

a
n

(
z
−

z
0

)
n

where

a
n

=

1

2
π
i

∮
γ

f
(
z
)

d

z

(
z
−

z
0

)

n
1

=

f

(
n
)

(

z
0

)

n
!

{\displaystyle f(z)=\sum\_{n=0}^\infty a\_n(z-z\_0)^n \quad \text{where} \quad a\_n=\frac{1}{2\pi i} \oint\_\gamma \frac{f(z)\,\mathrm{d}z}{(z-z\_0)^{n+1}} = \frac{f^{(n)}(z\_0)}{n!}}

The Laurent series for a complex function f(*z*) about a point *z0* is given by:

f
(
z

# n
−
∞

∞

a
n

(
z
−

z
0

)
n

where

a
n

=

1

2
π
i

∮
γ

f
(
z
)

d

z

(
z
−

z
0

)

n
1

=

f

(
n
)

(

z
0

)

n
!

{\displaystyle f(z)=\sum\_{n=-\infty}^\infty a\_n(z-z\_0)^n \quad \text{where} \quad a\_n=\frac{1}{2\pi i} \oint\_\gamma \frac{f(z)\,\mathrm{d}z}{(z-z\_0)^{n+1}} = \frac{f^{(n)}(z\_0)}{n!}}

The positive subscripts correspond to a line integral around the outer part of the annulus and the negative subscripts correspond to a line integral around the inner part of the annulus. In reality it makes no difference where the line integral is so both line integrals can be moved until they correspond to the same contour gamma. See also: Z-transform

The function

1

(
z
−
1
)
(
z
−
2
)

{\displaystyle \frac{1}{(z-1)(z-2)}}

 has poles at z=1 and z=2. It therefore has 3 different Laurent series centered on the origin (*z0* = 0):

For 0 < |z| < 1 the Laurent series has only positive subscripts and is the Taylor series.
For 1 < |z| < 2 the Laurent series has positive and negative subscripts.
For 2 < |z| the Laurent series has only negative subscripts.
Cauchy formula for repeated integration:

f

(
−
n
)

(
a
# )

1

(
n
−
1
)
!

∫
0
a

f
(
z
)

(

a
−
z

)

n
−
1

d

z

{\displaystyle f^{(-n)}(a) = \frac{1}{(n-1)!} \int\_0^a f(z) \left(a-z\right)^{n-1} \,\mathrm{d}z}

For every holomorphic functionw

f
(
z
# )
f
(
x
i
y
# )

f
x

(
x
,
y
)
i

f
y

(
x
,
y
)

{\displaystyle f(z)=f(x+iy)=f\_x(x,y)+i f\_y(x,y)}

 both fx and fy are harmonic functionsw.

Any two-dimensional harmonic function is the real part of a complex analytic functionw.

See also: complex analysisw.

fy is the harmonic conjugate of fx.
Geometrically fx and fy are related as having orthogonal trajectories, away from the zeroes of the underlying holomorphic function; the contours on which fx and fy are constant (equipotentials and streamlines) cross at right angles.
In this regard, fx+ify would be the complex potential, where fx is the potential function and fy is the stream function.
fx and fy are both solutions of Laplace's equationw

∇
2

# f
0

{\displaystyle \nabla^2 f = 0}

 so divergence of the gradient is zero
Legendre function are solutions to Legendre's differential equation.
This ordinary differential equation is frequently encountered when solving Laplace's equation (and related partial differential equations) in spherical coordinates.
A harmonic functionw is a scalar potential function therefore the curl of the gradient will also be zero.
See Potential theory

Harmonic functions are real analogues to holomorphic functions.
All harmonic functions are analytic, i.e. they can be locally expressed as power series.
This is a general fact about elliptic operators, of which the Laplacian is a major example.
The value of a harmonic function at any point inside a disk is a weighted average of the value of the function on the boundary of the disk.

P
[
u
]
(
x
# )

∫
S

u
(
ζ
)
P
(
x
,
ζ
)
d
σ
(
ζ
)
.

{\displaystyle P[u](x) = \int\_S u(\zeta)P(x,\zeta)d\sigma(\zeta).\,}

The Poisson kernel gives different weight to different points on the boundary except when x=0.
The value at the center of the disk (x=0) equals the average of the equally weighted values on the boundary. See The\_mean\_value\_property.
All locally integrable functions satisfying the mean-value property are both infinitely differentiable and harmonic.
The kernel itself appears to simply be 1/r^n shifted to the point x and multiplied by different constants.
For a circle (K = Poisson Kernel):

f
(
r

e

i
ϕ

# )

∮
0

2
π

f
(
R

e

i
θ

)
K
(
R
,
r
,
θ
−
ϕ
)
d
θ

{\displaystyle
f(re^{i\phi})=
\oint\_0^{2\pi} f(Re^{i\theta}) K(R,r,\theta-\phi) d\theta
}

d
(
a
(
x
,
y
)
i
b
(
x
,
y
)
)

d
(
x
i
y
)

=

d
a
i
d
b

d
x
i
d
y

=

(
d
a
i
d
b
)
(
d
x
−
i
d
y
)

d

x
2

d

y
2

=

d
a
d
x
d
b
d
y
i
(
d
b
d
x
−
d
a
d
y
)

d

x
2

d

y
2

{\displaystyle \frac{d(a(x,y)+ib(x,y))}{d(x+iy)} = \frac{da+idb}{dx+idy} = \frac{(da+idb)(dx-idy)}{dx^2+dy^2} = \frac{dadx+dbdy+i(dbdx-dady)}{dx^2+dy^2}}

d
(
a
(
x
,
y
)
i
b
(
x
,
y
)
)

d
(
x
i
y
)

=

d
a

d
x

d
b

d
y

i

(

d
b

d
x

−

d
a

d
y

)

=

∇
⋅
f

i

∇
×
f

{\displaystyle \frac{d(a(x,y)+ib(x,y))}{d(x+iy)} = \frac{da}{dx} +\frac{db}{dy} +i \bigg(\frac{db}{dx}- \frac{da}{dy} \bigg) =
{\color{red} \nabla \cdot f}
+ i {\color{blue} \nabla \times f}
}

1. ↑ Wikipedia:Residue (complex analysis)
2. ↑ Wikipedia:Potential theory
3. ↑ Wikipedia:Harmonic conjugate