Chapter 1 Complex Functions and Analytic Functions

Complex Numbers

An ordered pair of real numbers is a complex number.

z = x + iy = \rho{e^{i\phi}}

\phi is called the argument of z, denoted as

\phi = \arg(z)

The principal value of the argument \text{Arg}(z)

-\pi < \text{Arg}(z) \leq \pi

Regions

Neighborhood of a Point

|z - z_0| \leq \varepsilon_0

Interior Point

If z_0 and its neighborhood both belong to E, then z_0 is an interior point of E.

Boundary Point

Domain

Closure

Simply Connected Domain

Multiply Connected Domain

Complex Functions

Derivative

1. Differentiability implies continuity, but continuity does not imply differentiability.
2. If the real and imaginary parts are continuous, then the complex function is continuous.
3. If the real and imaginary parts are differentiable, the complex function is not necessarily differentiable.

Sufficient Conditions for Derivability

Cauchy-Riemann Equations

If w=f(z)=u(x,y) + iv(x,y) is differentiable at a point z = x+iy within a region D, then at (x,y) there exists and satisfies the Cauchy-Riemann equations

\frac{\partial{u}}{\partial{x}} = \frac{\partial{v}}{\partial{y}}, 
\frac{\partial{v}}{\partial{x}} = -\frac{\partial{u}}{\partial{y}}

In polar coordinates, it is

\frac{\partial{u}}{\partial{\rho}} = \frac{1}{\rho}\frac{\partial{v}}{\partial{\phi}}, 
\frac{\partial{v}}{\partial{\rho}} = -\frac{1}{\rho}\frac{\partial{u}}{\partial{\phi}} 

Analytic Functions

Definition: If a function f(z) is differentiable at every point in the neighborhood of z_0, then the function f(z) is said to be analytic at z_0. If f(z) is not analytic at z_0, then z_0 is called a singularity of the function f(z).

1. Analyticity implies differentiability. Being differentiable at a point and being analytic are not the same.
2. If the region D is open, then every point and its neighborhood belong to the region D, and differentiability and analyticity are equivalent within this region.

Properties of Analytic Functions

Orthogonality

If f(z) = u(x,y) + iv(x,y) is analytic in the region D, then the family of curves u(x,y)=C, v(x,y)=D is orthogonal in D.

That is

\frac{\partial{u}}{\partial{y}}\frac{\partial{v}}{\partial{y}} + \frac{\partial{u}}{\partial{x}}\frac{\partial{v}}{\partial{x}} = 0

Harmonic Functions

If f(z) = u(x,y) + iv(x,y) is analytic in the region D, then u, v are harmonic functions in the region D.

The real and imaginary parts are connected by the Cauchy-Riemann equations

Only the real or imaginary part of the analytic function is needed to determine the analytic function.

Conformal Mapping

Elementary Analytic Functions

Elementary Analytic Functions f(z)	Analytic Region	Derivative	Period	Single-Valued/Multi-Valued
z^n (n > 0 integer)	Entire plane	nz^{n-1}	Non-periodic function	Single-Valued
z^n (n < 0 integer)	Entire plane except z = 0	nz^{n-1}	Non-periodic function	Single-Valued
e^z	Entire plane	e^z	2\pi i	Single-Valued
\sin z	Entire plane	\cos z	2\pi	Single-Valued
\cos z	Entire plane	-\sin z	2\pi	Single-Valued
\sinh z	Entire plane	\cosh z	None	Single-Valued
\cosh z	Entire plane	\sinh z	None	Single-Valued
\ln z	Single-valued branch	\frac{1}{z}	Non-periodic function	Multi-Valued
z^s	Single-valued branch	z^z(\ln z + 1)	Non-periodic function	Multi-Valued

Chapter 2 Integrals of Analytic Functions

Integrals of Complex Functions

Closed Loop Integrals

\int_L f(z) dz = \int_L [u(x, y) dx - v(x, y) dy] + i\int_L [v(x, y) dx + u(x ,y) dy]

Properties

1. The modulus of the integral of a complex function is not greater than the integral of the modulus of the integrand

   | \int_L f(z) dz | \leq \int_L |f(z)| |dz| \leq \int_L |f(z)| dl

2. Estimation of Integrals
   If |f(z)| \leq M, then

   |\int_L f(z) dz| \leq Ml

Cauchy's Theorem

Cauchy's Theorem for Simply Connected Regions

If the function f(z) is analytic within and on the boundary line L of a simply connected region D, then the integral of f(z) along L or any closed curve l within D is 0

\oint_L f(z) dz = 0, \oint_l f(z) dz = 0

The integral of an analytic function is path-independent.

Cauchy's Theorem for Multiply Connected Regions

Let L be the outer boundary of a closed multiply connected region D, and C_1, C_2, \cdots, C_n be the inner boundaries, then

\oint_L f(z) dz + \sum_{k=1}^{n} \oint_{C_k}f(z)dz = 0

Theorem of Contour Deformation

Let C_1 and C_2 be two positively oriented simple closed curves, and C_1 is enclosed within C_2. If the function f is analytic in the multiply connected region enclosed by C_1 and C_2, then

\oint_{C_1} f(z) dz = \oint_{C_2} f(z) dz

An Important Integral

\oint_{C_R}\frac{dz}{(z-z_0)^n} = \begin{cases}2\pi i, n = 1\\0, n \neq 1 \end{cases}

C_R is a positively oriented circle with center z_0 and radius R.

Cauchy's Integral Formula for Simply Connected Regions

f(z_0) = \frac{1}{2\pi j}\oint_l \frac{f(z)}{ z - z_0} dz

where l is any positively oriented curve within D, and z_0 is any point within l.

Cauchy's Integral Formula for Multiply Connected Regions

f(z_0) = \frac{1}{2\pi j}\oint_{L + C_1 + C_2 + \cdots + C_n} \frac{f(z)}{z-z_0}dz

where L is the outer boundary, C_1, C_2, \cdots, C_n are the inner boundaries, and z_0 is any point within D.

Cauchy's Integral Formula for Unbounded Regions

If f(z) is analytic outside a closed curve L and f(z) \rightarrow 0 as |z| \rightarrow \infty, and z_0 is any point outside L, then

f(z_0) = \frac{1}{2\pi j}\oint_L \frac{f(z)}{z - z_0} dz

Some Corollaries

Derivatives

If f(z) is analytic within a simply connected region D, then its derivatives are also analytic within that region, and

f^{(n)}(z) = \frac{n!}{2\pi j}\oint_C \frac{f\xi}{(\xi - z)^{n+1}} dz ~~~~~~~n = 1, 2, 3, \cdots

Chapter 3 Series of Complex Functions

Properties of Complex Series

Theorem 1

The necessary and sufficient condition for the convergence of \sum_{k=1}^{\infty} w_k is: \forall \varepsilon > 0, \exists N \in \mathcal{N}, \text{such that} n > N \text{when}, |\sum_{k=n+1}^{n+p}w_k| < \varepsilon,

Theorem 2

Let w_k = u_k + iv_k, then the necessary and sufficient condition for the convergence of \sum_{k=1}^{\infty} w_k is that the real part \sum_{k=1}^{\infty} u_k and the imaginary part \sum_{k=1}^{\infty} v_k both converge.

Theorem 2'

Let w_k = u_k + iv_k, S = a + ib, then the necessary and sufficient condition for the convergence of \sum_{k=1}^{\infty} w_k to S is that the real part \sum_{k=1}^{\infty} u_k converges to a and the imaginary part \sum_{k=1}^{\infty} v_k converges to b.

Theorem 3

The necessary condition for the convergence of the series \sum_{k=1}^{\infty} w_k is \lim_{k \rightarrow \infty} w_k = 0

Theorem 4

If the series \sum_{k=1}^{\infty} |w_k| converges, then the series \sum_{k=1}^{\infty} w_k converges.

Theorem 5

The necessary and sufficient condition for the absolute convergence of \sum_{k=1}^{\infty} w_k is that \sum_{k=1}^{\infty} u_k and \sum_{k=1}^{\infty} v_k both converge absolutely.

Series of Complex Functions

Theorem 6

The necessary and sufficient condition for the convergence of the series of complex functions \sum_{n=0}^N f_n(z) is that for any point z in D, given any \varepsilon > 0, there exists N(z) such that when n > N(z), for any positive integer p, we have

|\sum_{k=n+1}^{n+p} f_k(z)| < \varepsilon

Uniform Convergence

\forall \varepsilon > 0, there exists an N independent of z, such that for all z on the region D or curve L: when n > N,

|\sum_{k=n+1}^{n+p} f_k(z)| < \varepsilon

Then the series \sum_{n=0}^N f_n(z) is said to converge uniformly on D or L.

Majorant Test

If there exists |f_n(z)| \leq M_n, and the series \sum_{n=1}^{\infty} M_n converges, then the series \sum_{n=1}^{\infty} f_n(z) converges absolutely and uniformly.

Power Series

Radius of Convergence

1. Ratio Test

   R = \lim_{n\rightarrow\infty}|\frac{a_n}{a_{n+1}}|

2. Root Test

   R = \lim_{n\rightarrow\infty}\frac{1}{|a_n|^{\frac{1}{n}}}

Properties

Property 1

Power series converge absolutely and uniformly within the circle of convergence.

Property 2

The sum, difference, and product of convergent series form a new series that converges within |z| < min(R_1, R_2).

Property 3

Let the radius of convergence of the power series be R, then its sum function is analytic within the circle of convergence, and can be differentiated or integrated term by term within its circle of convergence.

Taylor Series

Taylor Expansion of Analytic Functions

If f(z) is analytic within the region D: |z - z_0| < R, then for any point z in D, we have

f(z) = \sum_{k=0}^{\infty}a_k (z - z_0)^k, ~~|z - z_0| < R

where a_k = \frac{1}{2\pi i}\oint_C \frac{f(\xi)}{(\xi - z_0)^{k+1}}d\xi = \frac{f^{(k)}(z_0)}{k!} ~~~ k=0, 1, 2, \cdots

Radius of Convergence of Taylor Series

Theorem

If an analytic function is expanded into a Taylor series centered at z_0, then its circle of convergence is a circle centered at z_0 with a radius equal to the distance |z_0 - b| = R between z_0 and the nearest singularity b, and R is the radius of convergence of the Taylor series.

Laurent Series

A series of the form \sum_{n=-\infty}^{\infty}c_n (z - z_0)^n is called a Laurent series.

Theorem

If a function f(z) is analytic within the annular domain R_1 \leq |z - z_0| \leq R_2, then f(z) can be expanded into a Laurent series within this annular domain

f(z) = \sum_{n-\infty}^{\infty}c_n(z-z_0)^n

where the coefficients are

c_n = \frac{1}{2\pi i}\oint_C \frac{f(\xi)}{(\xi - z_0)^{n+1}}d\xi, ~~~n=0, \pm1,\pm2,\cdots

Convergence of Laurent Series

Let a,b be two adjacent singularities of f(z), and expand the function as a Laurent series centered at z_0 \sum_{n-\infty}^{\infty}c_n(z-z_0)^n, then the series converges within the annular domain |a - z_0| < |z - z_0| < |b - z_0|.

Isolated Singularities of Single-Valued Functions

If a function f(z) is not differentiable at a point z_0 but is continuously differentiable in any neighborhood of z_0 except at z_0, then z_0 is called an isolated singularity of f(z); if there are always non-differentiable points outside of z_0 in any neighborhood of z_0, then z_0 is called a non-isolated singularity of f(z).

On the punctured neighborhood of an isolated singularity, the single-valued analytic function f(z) can be expanded into a Laurent series \sum_{n-\infty}^{\infty}c_n(z-z_0)^n. The positive power part is the analytic part of the series, and the negative power part is the principal part of the series.

Three Types of Singularities

Removable Singularity

Any of the following can serve as a necessary and sufficient condition or definition for an isolated singularity

1. The Laurent series of f(z) in the neighborhood of the singularity z_0 has no principal part.

2. \lim_{z\rightarrow z_0} f(z) = c_0, c_0 \neq \infty

3. f(z) is bounded in the neighborhood of z_0

Pole of Order m

If the principal part of the Laurent series in the annular domain 0 < |z - z_0| < R is a finite number of terms, i.e.

f(z) = \sum_{n=-m}^{\infty}c_n (z-z_n)^n

then z_0 is called a pole of order m, and \lim_{z\rightarrow z_0}f(z) = 0

1. Laurent expansion
2. f(z) = \frac{1}{(z-z_0)^m \phi(z)}, \phi(z) is analytic and \phi(z_0) \neq 0
3. \lim_{z\rightarrow z_0} (z-z_0)^mf(z) = a ~(a \neq 0)

Essential Singularity

If the principal part of the Laurent expansion is an infinite number of terms, then z_0 is called an essential singularity.

1. The principal part of the Laurent expansion is infinite.
2. The limit \lim_{z\rightarrow z_0} f(z) does not exist.

Chapter 4 Residue Theorem and Its Applications

Residue Theorem

Residues at Finite Points

If the function f(z) is expanded into a Laurent series in the neighborhood of its singularity z_0

f(z) = \sum_{n=-\infty}^{\infty}c_n(z-z_0)^n

According to the conclusion

\oint_C \frac{1}{(z-z_0)^n}dz = \begin{cases}2\pi i, n = 1 \\0, ~~n\neq 0\end{cases}

Therefore

\oint_C f(z) dz = 2\pi i c_{-1}

c_{-1} is called the residue of f(z) at the finite point z_0, denoted as Resf(z_0) or Res[f(z), z_0] or Resf(z)|_{z=z_0}, that is

Resf(z_0) = \frac{1}{2\pi i}\oint_C f(z) dz = c_{-1}

The residue can be obtained by calculating the Laurent series or the contour integral.

Residue at Infinity

Resf(\infty) = \frac{1}{2\pi i}\oint_Lf(z)dz

where L is a closed contour that encircles z=0 in a clockwise direction and contains all the finite singularities of f(z).

At the same time, expand f(z) into a Laurent series in the neighborhood of z=\infty

f(z) = \sum_{k=-\infty}^{\infty}C_kz^k~~~~R < |z| < \infty

Integrate both sides along the contour L and use the conclusion from the previous section to have

\oint_L f(z) dz = C_{-1} 2\pi i

Therefore

Resf(\infty) = -C_{-1}

Theorem 1

If lim_{z\rightarrow \infty} f(z) = 0, then the residue at infinity

Resf(\infty)=-\lim_{z\rightarrow \infty} [z \cdot f(z)]

Theorem 2

If lim_{z\rightarrow \infty} f(z) \neq 0, then

Res[f(z), \infty] = -Res(f(\frac{1}{t}) \cdot \frac{1}{t^2}, 0)

Residue Theorem

Let the function f(z) be analytic in the region D except for a finite number of singularities z_1, z_2, \cdots, z_n, and let L be a positively oriented simple closed curve that encloses all the singularities, then

\oint_L f(z) dz = 2\pi i \sum_{k=1}^{n}Resf(z_k)

Residue Sum Theorem

Let the function f(z) be analytic on the extended complex plane except for a finite number of singularities z_k (k=1, 2, \cdots, n) and z=\infty, then

\sum_{k=1}^{n}Resf(z_k) + Resf(\infty) = 0

Calculation of Residues

1. When z_0 is a removable singularity Resf(z_0) = 0
2. When z_0 is an essential singularity, there is hardly any good method, only Laurent expansion or contour integration can be used.
3. When z_0 is a simple pole c_{-1} = \lim_{z\rightarrow z_0} (z-z_0)f(z)
   4. When z_0 is a pole of order m Resf(z_0) = \frac{1}{(m-1)!}\lim_{z\rightarrow z_0}\frac{d^{m-1}}{dz^{m-1}}[(z - z_0)^mf(z)]

Calculation of Real Integrals Using the Residue Theorem

Type \mathop{I} Real Integral Calculation

\int_0^{2\pi} R(cos\theta, sin\theta) d\theta = \oint_{|z|=1}R(\frac{z + z^{-1}}{2}, \frac{z - z^{-1}}{2i}) \frac{1}{iz} dz

Type \mathop{II} Real Integral Calculation

Let f(z) = \frac{P(z)}{Q(z)} be a rational function, where P(z) and Q(z) are coprime polynomials, and Q(z) has at least twice the degree of P(z), i.e., \lim_{z\rightarrow \infty} zf(z) = 0 and Q(z) has no zeros on the real axis, and there are a finite number of zeros in the upper half-plane, then

\int_{-\infty}^{+\infty}\frac{P(x)}{Q(x)} dx = 2\pi i \sum_{Imz_k > 0} Res[\frac{P(z)}{Q(z)}, z_k]

Type \mathop{III} Real Integral Calculation

Jordan's Lemma

Let C_R be the upper semicircle of |z|=R, and the function f(z) is continuous on C_R and \lim_{z\rightarrow \infty} f(z)=0, then

\lim_{|z|=R \rightarrow \infty} \int_{C_R}f(z)e^{iaz}dz = 0, ~~~a > 0

For the integral \int_{-\infty}^{+\infty} f(x)e^{iax}dx, ~(a > 0), if we take the function F(z)=f(z)e^{iaz}, and it satisfies 1. The function F(z)=f(z)e^{iaz} is analytic everywhere in the z-plane except for a finite number of singularities z_1, z_2, \cdots, z_n, and there are no singularities on the real axis 2. f(x) is a rational function, and the degree of the denominator is at least one higher than the degree of the numerator

\int_{-\infty}^{+\infty}f(x)e^{ia}dx = 2\pi i \sum_{Imz_k > 0} Res[f(z), z_k] 

Chapter 5 Fourier Series

Fourier Expansion of Periodic Functions

Fourier Series

Using the orthogonality of the trigonometric function family, we can obtain the Fourier series

f(x) = a_0 + \sum_{n=1}^{\infty}a_ncos\frac{n\pi x}{l} + b_nsin\frac{n\pi x}{l} = a_0 + \sum_{n=1}^{\infty}a_ncos\frac{2n\pi x}{T} + b_nsin\frac{2n\pi x}{T}

where

\begin{aligned}
\begin{cases}
a_0 = \frac{1}{2l} \int_{-l}^{l} f(x) dx = \frac{1}{T} \int_{-T}^{T} f(x) dx\\ \\
a_n = \frac{1}{l} \int_{-l}^{l}f(x)cos(\frac{n \pi x}{l}) dx = \frac{2}{T} \int_{-T}^{T}f(x)cos(\frac{2n \pi x}{T}) dx \\\\
b_n = \frac{1}{l} \int_{-l}^{l}f(x)sin(\frac{n \pi x}{l}) dx = \frac{2}{T} \int_{-T}^{T}f(x)sin(\frac{2n \pi x}{T}) dx\\\\
\end{cases}
\end{aligned}

Convergence of Fourier Series

Dirichlet's Theorem

If the function f(x) is continuous everywhere, or has only a finite number of first-kind discontinuities in each period; there are only a finite number of extreme points in each period, then the series converges, and at the convergence points, we have

a_0 + \sum_{n=1}^{\infty}a_ncos\frac{n\pi x}{l} + b_nsin\frac{n\pi x}{l} = f(x)

At the points of discontinuity, we have

a_0 + \sum_{n=1}^{\infty}a_ncos\frac{n\pi x}{l} + b_nsin\frac{n\pi x}{l} = \frac{1}{2}[f(x + 0) + f(x - 0)]

Fourier Expansion of Odd and Even Functions

Fourier Expansion of Bounded Functions

• Odd extension
• Even extension
• Periodic extension

Complex Form of Fourier Series

Take the complex exponential orthogonal function family

\cdots, e^{-i\frac{k\pi x}{l}}, \cdots, e^{-i\frac{2\pi x}{l}}, e^{-i\frac{\pi x}{l}}, 1, e^{i\frac{1\pi x}{l}}, e^{i\frac{2\pi x}{l}}, \cdots, e^{i\frac{k\pi x}{l}}, \cdots

It is not difficult to derive the generalized Fourier series based on this function family

f(x) = \sum_{k=-\infty}^{\infty}C_ke^{i\frac{k\pi x}{l}}

where

C_k = \frac{1}{2l}\int_{-l}^{l}f(x)(e^{i\frac{k\pi x}{l}})^{*} dx = \frac{1}{2l}\int_{-l}^{l}f(x)e^{-i\frac{k\pi x}{l}} dx

It can be seen that C_{-k} = C^*_k, |C_k| = \frac{\sqrt{a_k^2 + b_k^2}}{2}

Chapter 6 Mathematical Physics Boundary Value Problems

Typical Mathematical Equations

Wave Equation

u_{tt} - a^2\nabla^2u = f(x, y, z, t) 

\nabla^2A = \nabla \times \nabla \times A + \nabla(\nabla \cdot A)

Heat Conduction Equation

u_t -D\nabla^2u = f(x, y, z, t)

Poisson Equation

\nabla^2u = f(x, y, z, t)

Boundary Conditions

Initial Conditions

Boundary Conditions

Dirichlet Condition

u(x, y, z, t) |_{x_0, y_0, z_0} = f(x_0, y_0, z_0, t)

Neumann Condition

\frac{\partial \vec u}{\partial \vec n}|_{x_0, y_0, z_0} = f(x_0, y_0, z_0, t)

Mixed Boundary Condition

(u + H\frac{\partial \vec u}{\partial \vec n})|_{x_0, y_0, z_0} = f(x_0, y_0, z_0, t)

Natural Boundary Condition

Second-Order Linear Partial Differential Equations

A(x, y)\frac{\partial^2u}{\partial x^2} + B(x,y)\frac{\partial^2u}{\partial x \partial y} + C(x, y)\frac{\partial^2u}{\partial y^2} + D(x, y)\frac{\partial u}{\partial x} + E(x,y)\frac{\partial u}{\partial y} + F(x, y)u = G(x,y)

Take the discriminant \Delta = B(x, y)^2 - 4A(x,y)C(x,y)

If \Delta > 0 the equation is hyperbolic, typically the wave equation u_{tt} - a^2u_{xx} = 0

If \Delta < 0 the equation is elliptic, typically the Laplace equation \nabla^2u=0

If \Delta = 0 the equation is parabolic, typically the transport equation u_t - a^2u_{xx} = 0

Linear partial differential equations can use the principle of superposition.

Method of Traveling Waves and d'Alembert's Formula

d'Alembert's Formula

For the boundary value problem

\begin{aligned}
\begin{cases}
u_{tt} - a^2u_{xx} = 0 \\
u(x, 0) = \varphi (x) \\
u_t(x, 0) = \psi (x) \\
\end{cases}
\end{aligned}

The equation can be written as

(\frac{\partial}{\partial t} + a\frac{\partial}{\partial x})(\frac{\partial}{\partial t} - a\frac{\partial}{\partial x})u = 0

After transformation

\xi = x + at ~~~~~~
\eta = x - at

It is not difficult to solve

u(x, t) = \frac{1}{2}[\varphi(x + at) + \varphi(x - at)] + \frac{1}{2a}\int_{x-at}^{x+at} \psi (\xi) d\xi

This is the solution to the one-dimensional wave equation under given initial conditions, known as d'Alembert's formula.

D'Alembert's formula is only applicable to homogeneous wave equation problems without boundaries in space.

Chapter 7 Method of Separation of Variables

Conditions for Separation of Variables

• Constant coefficient second-order homogeneous partial differential equations can always separate variables.
• Variable coefficient second-order homogeneous partial differential equations need to meet certain conditions.
• Boundary conditions need to be homogeneous.

General Steps of the Method of Separation of Variables

Variable Separation

Solve the Eigenvalue Problem

Find Particular Solutions and Superpose to Obtain the General Solution

Determine the Undetermined Coefficients Through Non-Homogeneous Initial or Boundary Conditions

General Methods for Homogenizing Non-Homogeneous Boundary Conditions

For the boundary value problem

\begin{aligned}
\begin{cases}
\frac{\partial^2u}{\partial t^2} - a^2\frac{\partial^2 u}{\partial x^2} = f(x, t) , ~~~~~~~~~~~~~~~~~~~~~~0 < x < l, t > 0\\\\
u(0, t) = g_1(t), ~u(l, t) = g_2(t), ~~~~~~~~~ t > 0 \\\\
u(x, 0) = \phi(x), ~u_t(x, 0) = \psi(x), ~~~~~ 0 < x < l\\
\end{cases}
\end{aligned}

Using the principle of superposition, let

u(x, t) = v(x, t) + w(x, t)

We can choose

w(x, t) = \frac{x}{l}[g_2(t) - g_1(t)] + g_1(t)

Then the boundary value problem for v(x, t)

\begin{aligned}
\begin{cases}
\frac{\partial^2v}{\partial t^2} - a^2\frac{\partial^2 v}{\partial x^2} = f(x, t) - \frac{x}{l}[g''_2(t) - g''_1(t)] - g''_1(t) \\\\
v(0, t) 0, ~v(l, t) = 0 \\\\
v(x, 0) = \phi(x) - \frac{x}{l}[g_2(0) - g_1(0)] - g_1(0) \\\\
v_t(x, 0) = \psi(x) - \frac{x}{l}[g'_2(0) - g'_1(0)] - g'_1(0)
\end{cases}
\end{aligned}

Common Eigenvalues and Eigenfunctions of Eigenvalue Problems

Boundary Conditions	Eigenvalues \lambda	Eigenfunctions X(x)
X\lvert_{x=0} = 0, X\lvert_{x=l} = 0	\lambda = \left( \frac{n \pi}{l} \right)^2, n=1,2,3,\ldots	\sin \left( \frac{n \pi}{l} x \right)
X'\lvert_{x=0} = 0, X'\lvert_{x=l} = 0	\lambda = \left( \frac{n \pi}{l} \right)^2, n=0,1,2,\ldots	\cos \left( \frac{n \pi}{l} x \right)
X\lvert_{x=0} = 0, X'\lvert_{x=l} = 0	\lambda = \left( \frac{(n+\frac{1}{2}) \pi}{l} \right)^2, n=0,1,2,\ldots	\sin \left( \frac{(n+\frac{1}{2}) \pi}{l} x \right)
X'\lvert_{x=0} = 0, X\lvert_{x=l} = 0	\lambda = \left( \frac{(n+\frac{1}{2}) \pi}{l} \right)^2, n=0,1,2,\ldots	\cos \left( \frac{(n+\frac{1}{2}) \pi}{l} x \right)

Chapter 8 Series Solution and Eigenvalue Problem of Second-Order Ordinary Differential Equations

Separation of Variables in Common Orthogonal Coordinate Systems

Separation of Variables in Laplace's Equation

Spherical Coordinate System

\frac{1}{r^2}\frac{\partial}{\partial r}(r^2\frac{\partial u}{\partial r}) + \frac{1}{r^2sin^2\theta}(sin\theta \frac{\partial u}{\partial \theta}) + \frac{1}

{r^2sin^2\theta}\frac{\partial^2 u}{\partial \varphi^2} = 0

The solution to the equation is

\Phi_m = A_msinm\varphi + B_mcosm\varphi

R(r) = Cr^l + \frac{D}{r^{l+1}}

(1-x^2)\frac{d^2\Theta}{dx^2} - 2x\frac{d\Theta}{dx} + [l(l+1) - \frac{m^2}{1-x^2}]\Theta = 0

The last equation is the l-th order associated Legendre equation, and for all physical problems, \Theta(\theta) satisfies the following eigenvalue problem:

\begin{aligned}

\begin{cases}

(1-x^2)\frac{d^2\Theta}{dx^2} - 2x\frac{d\Theta}{dx} + [l(l+1) - \frac{m^2}{1-x^2}]\Theta = 0 \\\\

\Theta(0) = \text{finite}, ~~\Theta(\pi) = \text{finite}

\end{cases}

\end{aligned}

Cylindrical Coordinate System

\frac{1}{\rho}\frac{\partial}{\partial \rho}(\rho \frac{\partial u}{\partial \rho}) + \frac{1}{\rho^2}\frac{\partial^2 u}{\partial \varphi^2} + \frac{\partial^2u}{\partial z^2} = 0

The solution for \Phi is

\Phi(\varphi) = A_msinm\varphi + B_mcosm\varphi

When \mu = 0, the solution is

\begin{aligned}

\begin{cases}

R(\rho) = \begin{cases}A_0 + B_0ln\rho ,~~m=0\\\\A_m\rho^m + B_m\rho^{-m}, ~~ m\neq 0\end{cases}\\\\

Z(z) = C + Dz

\end{cases}

\end{aligned}

When \mu > 0, the solution is

Z(z) = Ce^{\sqrt{u}z} + De^{-\sqrt{u}z}

The equation for R is

x^2\frac{d^2R}{dx^2} + x\frac{dR}{dx} + (x^2 - m^2)R = 0

This is the standard Bessel equation.

When \mu < 0, the solution is

Z(z) = Csin{\sqrt{-\mu}z} + Dcos{\sqrt{-\mu}z}

The equation for R is

x^2\frac{d^2R}{dx^2} + x\frac{dR}{dx} + (-x^2 - m^2)R = 0

This is the standard imaginary Bessel equation.

Separation of Variables in the Wave Equation

u_{tt}(\vec{r}, t) - a^2\nabla^2u(\vec{r}, t) = 0

The result is

\begin{aligned}

\begin{cases}

T''(t) + k^2a^2T(t) = 0 \\\\

\nabla^2V(\vec{r}) + k^2V(\vec{r}) = 0

\end{cases}

\end{aligned}

Separation of Variables in the Heat Conduction Equation

u_{t}(\vec{r}, t) - a^2\nabla^2u(\vec{r}, t) = 0

The result is

\begin{aligned}

\begin{cases}

T'(t) + k^2a^2T(t) = 0 \\\\

\nabla^2V(\vec{r}) + k^2V(\vec{r}) = 0

\end{cases}

\end{aligned}

Separation of Variables in the Helmholtz Equation

Spherical Coordinate System

\frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial \Psi}{\partial r} \right) + \frac{1}{r^2 \sin\theta} \frac{\partial}{\partial \theta} \left( \sin\theta \frac{\partial \Psi}{\partial \theta} \right) + \frac{1}{r^2 \sin^2\theta} \frac{\partial^2 \Psi}{\partial \phi^2} + k^2 \Psi = 0

The solution is

\Phi_m = A_msinm\varphi + B_mcosm\varphi

The equation for R is

r^2\frac{d^2R}{dr^2} + 2r\frac{dR}{dr} + [k^2r^2 - l(l+1)]R = 0

Let x = kr, R(r) = \sqrt{\frac{\pi}{2x}}y(x)

x^2\frac{d^2y}{dx^2} + x\frac{dy}{dx} + (x^2 - (l + \frac{1}{2})^2)y = 0

Cylindrical Coordinate System

\frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial \Psi}{\partial r} \right)+ \frac{1}{r^2} \frac{\partial^2 \Psi}{\partial \phi^2}+ \frac{\partial^2 \Psi}{\partial z^2}+ k^2 \Psi = 0

Series Solution in the Neighborhood of Regular Points

Regular and Singular Points of the Equation

For the equation

\frac{d^2w(z)}{dz^2} + p(z)\frac{dw(z)}{dz} + q(z)w(z) = 0

If the functions p(z) and q(z) are analytic at z_0, then z_0 is a regular point of the equation. If z_0 is a singular point of the equation, then z_0 is a singular point of the equation. If z_0 is a singular point of p(z) of no higher than the first order, and q(z) is no higher than the second order, then z_0 is called a regular singular point of the equation.

Power Series Solution in the Neighborhood of a Regular Point

w(z) = \sum_{k=0}^{\infty}a_k(z-z_0)^k

Sturm-Liouville Eigenvalue Problem

The Sturm-Liouville Equation

\begin{aligned}

\frac{d}{dx}[k(x)\frac{dy}{dx}] - q(x)y + \lambda P(x)y = 0 \\\\

k(x) \geq 0, ~~q(x) \geq 0, ~~P(x) \geq 0, ~~ a\leq x \leq b

\end{aligned}

Chapter 9 Special Functions

Legendre Polynomials

Series Solution of the Legendre Equation

The Legendre equation in spherical coordinates is:

\frac{1}{\sin\theta}\frac{d}{d\theta}(\sin\theta \frac{d\Theta}{d\theta}) + [l(l + 1) - \frac{m^2}{\sin^2\theta}]\Theta = 0

This equation can be written as:

(1-x^2)\frac{d^2y}{dx^2} - 2x\frac{dy}{dx} + [l(l+1) - \frac{m^2}{1-x^2}]y = 0

The solution to the Legendre equation is the Legendre series:

P_l(x) = \sum_{k=0}^{[\frac{l}{2}]}(-1)^k\frac{(2l - 2k)!}{2^lk!(l-k)!(l-2k)!}x^{l-2k}

The first few terms are:

P_0(x) = 1

P_1(x) = x

P_2(x) = \frac{1}{2}(3x^2 - 1)

P_3(x) = \frac{1}{2}(5x^3 - 3x)

P_4(x) = \frac{1}{8}(35x^4 - 30x^2 + 3)

Properties of Legendre Polynomials

Parity

P_l(-x) = (-1)^lP_l(x)

Special Values

P_{2n+1}(0)=0

P_{2n} = (-1)^n\frac{(2n-1)!!}{(2n)!!}

P_l(1) = 1, P_l(-1) = (-1)^l

Orthogonality

Eigenfunctions of the S-L problem are weighted orthogonal.

Differential and Integral Representations of Legendre Polynomials

P_l(x) = \frac{1}{2^ll!}\frac{d^l}{dx^l}(x^2 - 1)^l

P_l(x) = \frac{1}{2\pi i}\frac{1}{2^l}\oint_C\frac{(\xi^2-1)^l}{(\xi-x)^{l+1}}dx

P_l^m(x) = \frac{(1-x^2)^{\frac{m}{2}}}{2^ll!}\frac{d^{l+m}}{dx^{l+m}}(x^2-1)^l

Boundary Value Problem

Solution of Laplace's Equation in Spherical Coordinates

\begin{aligned}
\begin{cases}
R_l(r) = C_lr^l + \frac{D_l}{r^{l+1}} \\\\
\Phi_m(\varphi) = Asinm\varphi + Bcosm\varphi \\\\
\Theta_{m, l}(\theta) = P_l^m(\cos\theta)
\end{cases}
\end{aligned}

Recurrence Relations for Legendre Polynomials

From the generating function for Legendre polynomials, we derive the following recurrence relation:

(n+1)P_{n+1}(x) - (2n+1)xP_n(x) + nP_{n-1}(x) = 0 

P_n(x) = P'_{n+1}(x) - 2xP'_n(x) + P'_{n-1}(x), ~~ n\geq1

From these two formulas, we can derive:

(2n+1)P_n(x) = P'_{n+1}(x) - P'_{n-1}(x), ~~n \geq 1

P'_{n+1}(x) = (n+1)P_n(x) + xP'_n(x), ~~~n \geq 0

nP_n(x) = xP'_n(x) - P'_{n-1}(x), ~~~n\geq1

(x^2-1)P'_n(x) = nxP(x) - nP_{n-1}(x), ~~n\geq1

Bessel Functions

Three Types of Cylindrical Functions

The Bessel equation is:

x^2\frac{d^2y}{dx^2} + x\frac{dy}{dx} + (x^2 - \nu^2)y = 0

The general solution to the Bessel equation is:

y(x) = c_1y_1(x) + c_2y_2(x)

where

y_1(x) = \sum_{n=0}^{\infty}\frac{(-1)^nc_0\Gamma(\nu + 1)}{2^{2n}n!\Gamma(\nu + n + 1)}

y_2(x) = \sum_{n=0}^{\infty}\frac{(-1)^nc_0\Gamma(-\nu + 1)}{2^{2n}n!\Gamma(-\nu + n + 1)}

By choosing the constant c_0 = \frac{1}{2^{\nu}\Gamma(\nu + 1)} in y_1(x), we define y_1(x) as J_\nu(x), known as the \nuth-order Bessel function:

J_\nu(x) = \sum_{n=0}^{\infty}\frac{(-1)}{n!\Gamma(\nu + n + 1)}\left(\frac{x}{2}\right)^{2n+\nu}

By choosing the constant c_0 = \frac{1}{2^{-\nu}\Gamma(\nu + 1)} in y_2(x), we define y_2(x) as J_{-\nu}(x), known as the \nuth-order Bessel function:

J_{-\nu}(x) = \sum_{n=0}^{\infty}\frac{(-1)}{n!\Gamma(-\nu + n + 1)}\left(\frac{x}{2}\right)^{2n-\nu}

J_\nu(x) and J_{-\nu}(x) are called the first kind cylindrical functions. When \nu is not an integer, they are linearly independent. In this case, the general solution is:

y(x) = A_\nu J_\nu(x) + B_\nu J_{-\nu}(x)

When \nu is an integer, the second general solution is:

y(x) = A_\nu J_\nu(x) + B_\nu N_{\nu}(x)

where

N_{\nu}(x) = \frac{\cos\nu\pi J_{\nu}(x) - J_{-\nu}(x)}{\sin\nu \pi}

N_{\nu}(x) is called the second kind cylindrical function, also known as the Neumann function.

The third form of the general solution is:

y(x) = A_\nu H_\nu^{(1)}(x) + B_\nu H_{\nu}^{(2)}(x)

where

\begin{aligned}
\begin{cases}
H_{\nu}^{(1)}(x) = J_{\nu}(x) + iN_{\nu}(x)\\\\
H_{\nu}^{(2)}(x) = J_{\nu}(x) - iN_{\nu}(x)
\end{cases}
\end{aligned}

These are called the third kind cylindrical functions, also known as Hankel functions.

Properties of Bessel Functions

Values at Special Points

\begin{aligned}
\lim_{x\rightarrow0}J_0(x) = 1, \lim_{x\rightarrow0}J_\nu(x) = 0, \lim_{x\rightarrow0}J_{-\nu}(x) = \infty, \lim_{x\rightarrow0}N_\nu(x) = \infty, \\\\
\lim_{x\rightarrow\infty}J_\nu(x) = 0, \lim_{x\rightarrow\infty}J_{-\nu}(x) = 0, \lim_{x\rightarrow\infty}N_\nu(x) = 0
\end{aligned}

Recurrence Relations for Bessel Functions

\frac{d}{dx}[x^vZ_v(x)] = x^vZ_{v-1}(x)

\frac{d}{dx}[x^{-v}Z_v(x)] = -x^{-v}Z_{v+1}(x)

Common Corollaries

J'_0(x) = -J_1(x)

[xJ_1(x)]' = xJ_0(x)

xJ_1(x) = \int_0^x\xi J_0(\xi) d\xi