$$一个伟大的数学家大概需要奋斗e^{\pi}年$$
$$e^{i\pi} + 1 = 0$$
$$e^{\pi}=23.140692632779269005729086367949$$
$$365\times e^{\pi} = 8446.3528109644331870911165243012$$
$$天赋-伟大=365\times e^{\left( i+1 \right)\pi}个昼夜$$

## 什么是数学美？

reference：Golden ratio

$$黄金分割点：\frac{\sqrt{5} -1}{2} =0.61803398874989484820458683436564$$

（LaTeX，音译“拉泰赫”）是一种基于ΤΕΧ的排版系统，由美国计算机学家莱斯利·兰伯特（Leslie Lamport）在20世纪80年代初期开发，利用这种格式，即使使用者没有排版和程序设计的知识也可以充分发挥由TeX所提供的强大功能，能在几天，甚至几小时内生成很多具有书籍质量的印刷品。对于生成复杂表格和数学公式，这一点表现得尤为突出。因此它非常适用于生成高印刷质量的科技和数学类文档。这个系统同样适用于生成从简单的信件到完整书籍的所有其它种类的文档。

reference： LaTeX

### 蝴蝶曲线

polar equation:

$$r =e^{cos\theta } -2cos4\theta +\frac{sin^{5}\theta }{12} r=e^{\sin \theta }-2\cos4\theta +\sin ^{5}\left({\frac {2\theta -\pi }{24}}\right)$$

parametric equations:

$$x=\sin t \left(e^{\cos t }-2\cos 4t -\sin ^{5}\left({t \over 12}\right)\right)y=\cos t \left(e^{\cos t}-2\cos 4t -\sin ^{5}\left({t \over 12}\right)\right)$$

reference：Butterfly curve (transcendental) )

### 心形线

$$r =a(1-cos\theta )$$

$$x^2+y^2+ax=a \sqrt{ x^2+y^2 }$$

$$r =a(1+cos\theta )$$

$$x^2+y^2-ax=a \sqrt{ x^2+y^2 }$$

$$r =a(1-sin\theta )$$

$$r =a(1+sin\theta )$$

$$x=a(2cost-cos2t)$$

$$y=a(2sint-sin2t)$$

reference：Cardioid

### 笛卡尔叶形线

$$x^3+y^3=3axy$$

$$x=\frac{3at}{1+t^3}$$

$$y=\frac{3at^2}{1+t^3}$$

reference：Folium of Descartes

### 斐波那契螺线

$$F_{0} =0$$

$$F_{1} =1$$

$$F_{n} =F_{n-1}+F_{n-2}$$

reference：Fibonacci number 发现什么了吗？······

Golden spiral

Spiral

### 对数螺线

$$r=e^{\alpha \theta }$$

In parametric form, the curve is:

$$x(t) = r(t) \cos t = e^{\alpha \theta } \cos t\,y(t) = r(t) \sin t = e^{\alpha \theta } \sin t\,$$

reference：Logarithmic spiral

Spiral

### 阿基米德螺线

$$r=a\theta$$

or , it can be described by the equation:

$$r=a+b\theta$$

reference： Archimedean spiral

Spiral

The involute of a circle (black) is not identical to the Archimedean spiral (red).

reference： Spiral

### 双曲螺线

$$r\theta =a$$

reference：Hyperbolic spiral

Spiral

### 费马螺线

$$r^{2} =a^{2} \theta$$

reference：Fermat’s spiral

Spiral

### Euler spiral

reference：Euler spiral

Spiral

### lituus

reference：Lituus (mathematics))

Spiral

### spiral of Theodorus

$$\Delta r={\sqrt {n+1}}-{\sqrt {n}}$$

reference：Spiral of Theodorus

Theodorus of Cyrene

Spiral

### 伯努利双纽线

$$(x^2+y^2)^2=a^2(x^2-y^2)$$

$$r^2=a^2cos2\theta$$

As a parametric equation:

$$x={\frac {a\cos t}{\sin ^{2}t+1}}; y={\frac{ a \cos t\sin t}{\sin ^{2} t+1}}$$

$$(x^2+y^2)^2=2a^2xy$$

$$r^2=a^2sin2\theta$$

reference： Lemniscate of Bernoulli

reference related articles：

### A quartic plane curve

A quartic plane curve is a plane algebraic curve of the fourth degree. It can be defined by a bivariate quartic equation:

$$Ax^4+By^4+Cx^3y+Dx^2y^2+Exy^3+Fx^3+Gy^3+Hx^2y+Ixy^2+Jx^2+Ky^2+Lxy+Mx+Ny+P=0$$

reference： Quartic plane curve

### Ampersand curve

reference： Quartic plane curve

### Bicuspid curve

The biscuspid is a quartic plane curve with the equation:

$$(x^{2}-a^{2})(x-a)^{2}+(y^{2}-a^{2})^{2}=0$$

reference：Quartic plane curve

### 三叶玫瑰线

$$r=a cos3\theta$$

The three-leaved clover is the quartic plane curve.

$$x^4+2x^2y^2+y^4-x^3+3xy^2=0$$

$$r=a sin3\theta$$

reference：Quartic plane curve

### 四叶玫瑰线

polar equation:

$$r=a sin2\theta$$

with corresponding algebraic equation:

$$(x^2+y^2)^3 = 4x^2y^2$$

polar equation:

$$r=a cos2\theta$$

with corresponding algebraic equation:

$$(x^2+y^2)^3 = (x^2-y^2)^2$$

### rose curve

these curves can all be expressed by a polar equation of the form:

$$r=\cos(k\theta ) and r=\sin(k\theta )$$

reference related articles：Rose (mathematics))

### Maurer rose

Let $r = sin(n\theta )$ be a rose in the polar coordinate system, where n is a positive integer. The rose has n petals if n is odd, and 2n petals if n is even.

We then take 361 points on the rose:

$$(sin(nk), k) (k = 0, d, 2d, 3d, …, 360d),$$

where d is a positive integer and the angles are in degrees, not radians.

A Maurer rose of the rose $r = sin(n\theta )$ consists of the 360 lines successively connecting the above 361 points. Thus a Maurer rose is a polygonal curve with vertices on a rose.

Explanation:A Maurer rose can be described as a closed route in the polar plane. A walker starts a journey from the origin, (0, 0), and walks along a line to the point $(sin(nd), d)$. Then, in the second leg of the journey, the walker walks along a line to the next point, $(sin(n·2d), 2d)$, and so on. Finally, in the final leg of the journey, the walker walks along a line, from $(sin(n·359d), 359d)$ to the ending point, $(sin(n·360d), 360d)$. The whole route is the Maurer rose of the rose $r = sin(n\theta )$. A Maurer rose is a closed curve since the starting point, (0, 0) and the ending point, $(sin(n·360d), 360d)$, coincide.

The following figure shows the evolution of a Maurer rose (n = 2, d = 29° ).

The following are some Maurer roses drawn with some values for n and d:

reference： Maurer rose)

reference related articles：

Rose (mathematics)

Dual curve

Quatrefoil

### 反雪花曲线

$$A_{n} =A_{n-1} -\frac{\left(\frac{4}{3} \right) ^{n-1} }{3^n}$$

### 雪花曲线——Koch snowflake

The Koch snowflake (also known as the Koch curve, Koch star, or Koch island).

After each iteration, the number of sides of the Koch snowflake increases by a factor of 4, so the number of sides after n iterations is given by:

$$N_{n}=N_{n-1}\cdot 4=3\cdot 4^{n}$$

If the original equilateral triangle has sides of length s , the length of each side of the snowflake after n iterations is:

$$S_{n}={\frac {S_{n-1}}{3}}={\frac {s}{3^{n}}}$$

the perimeter of the snowflake after n iterations is:

$$P_{n}=N_{n}\cdot S_{n}=3\cdot s\cdot {\left({\frac {4}{3}}\right)}^{n}$$

reference： Koch snowflake

Variants of the Koch curve , for example:

1D, 90° angle

The first 2 iterations. Its fractal dimension equals 3/2 and is exactly half-way between dimension 1 and 2. It is therefore often chosen when studying the physical properties of non-integer fractal objects.

### 椭球面

$$\frac{x^{2}}{a^2}+\frac{y^{2}}{b^2}+\frac{z^{2}}{c^2} =1$$

reference：Ellipsoid

### 单叶双曲面——Hyperboloid of one sheet

$$\frac{x^{2}}{a^2}+\frac{y^{2}}{b^2}-\frac{z^{2}}{c^2} =1$$

### 双叶双曲面——Hyperboloid of two sheets

$$\frac{x^{2}}{a^2}+\frac{y^{2}}{b^2}-\frac{z^{2}}{c^2} =-1$$

reference： Hyperboloid

Paraboloid of revolution

$$z={\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}$$

$$kz=x^{2}+y^{2}$$

reference： Paraboloid

### 双曲抛物面，又称马鞍面——Hyperbolic paraboloid

$$-\frac{x^{2}}{2a}+\frac{y^{2}}{2b} =z( ab>0)$$

reference： Paraboloid

reference：Pythagoras tree (fractal))

$$E=m{c^{2} }$$

reference：Mass–energy equivalence

In quantum mechanics, the Schrödinger equation is a partial differential equation that describes how the quantum state of a quantum system changes with time.

The concept of a wavefunction is a fundamental postulate of quantum mechanics.

Time-dependent Schrödinger equation (general)

The form of the Schrödinger equation depends on the physical situation (see below for special cases). The most general form is the time-dependent Schrödinger equation, which gives a description of a system evolving with time.

$$i\hbar {\frac {\partial }{\partial t}}\Psi (\mathbf {r} ,t)={\hat {H}}\Psi (\mathbf {r} ,t)$$

$\hat{H}$ is the Hamiltonian operator (which characterizes the total energy of any given wave function and takes different forms depending on the situation).

reference：Hamiltonian (quantum mechanics) )

Time-dependent Schrödinger equation (single non-relativistic particle)

The most famous example is the non-relativistic Schrödinger equation for a single particle moving in an electric field (but not a magnetic field; see the Pauli equation)——Pauli equation

$$i\hbar {\frac {\partial }{\partial t}}\Psi (\mathbf {r} ,t)=\left[{\frac {-\hbar ^{2}}{2\mu }}\nabla ^{2}+V(\mathbf {r} ,t)\right]\Psi (\mathbf {r} ,t)$$

The term “Schrödinger equation” can refer to both the general equation (first box above), or the specific nonrelativistic version (second box above and variations thereof). The general equation is indeed quite general, used throughout quantum mechanics, for everything from the Dirac equation to quantum field theory, by plugging in various complicated expressions for the Hamiltonian. The specific nonrelativistic version is a simplified approximation to reality, which is quite accurate in many situations, but very inaccurate in others (see relativistic quantum mechanics and relativistic quantum field theory).

The time-dependent Schrödinger equation described above predicts that wave functions can form standing waves, called stationary states (also called “orbitals”, as in atomic orbitals or molecular orbitals). These states are important in their own right, and if the stationary states are classified and understood, then it becomes easier to solve the time-dependent Schrödinger equation for any state. Stationary states can also be described by a simpler form of the Schrödinger equation, the time-independent Schrödinger equation. (This is only used when the Hamiltonian itself is not dependent on time explicitly. However, even in this case the total wave function still has a time dependency.)

Time-independent Schrödinger equation (general)

$${\displaystyle \operatorname {\hat {H}} \Psi =E\Psi }$$

Time-independent Schrödinger equation (single non-relativistic particle)

As before, the most famous manifestation is the non-relativistic Schrödinger equation for a single particle moving in an electric field (but not a magnetic field):

$${\displaystyle \left[{\frac {-\hbar ^{2}}{2\mu }}\nabla ^{2}+V(\mathbf {r} )\right]\Psi (\mathbf {r} )=E\Psi (\mathbf {r} )}$$

reference：Schrödinger equation

### 波函数

reference： Wave function

### 不确定性原理（测不准原理）

$\hbar$ is the reduced Planck constant,$\hbar=\frac{h}{2 \pi}$.

$\hbar$ 是约化普朗克常数，又称合理化普朗克常数，是角动量的最小衡量单位，$\hbar=\frac{h}{2 \pi}$.

The most common general form of the uncertainty principle is the Robertson uncertainty relation:
$${\displaystyle \sigma _{A}\sigma _{B}\geq \left|{\frac {1}{2i}}\langle [{\hat {A}},{\hat {B}}]\rangle \right|={\frac {1}{2}}\left|\langle [{\hat {A}},{\hat {B}}]\rangle \right|}$$
The Robertson uncertainty relation immediately follows from a slightly stronger inequality, the Schrödinger uncertainty relation:
$${\displaystyle \sigma _{A}^{2}\sigma _{B}^{2}\geq \left|{\frac {1}{2}}\langle[ \hat A,\hat B]\rangle -\langle {\hat {A}}\rangle \langle {\hat {B}}\rangle \right|^{2}+\left|{\frac {1}{2i}}\langle [{\hat {A}},{\hat {B}}]\rangle \right|^{2}}$$

reference：Uncertainty principle

### 自旋

As a solution for a certain partial differential equation, the quantized angular momentum can be written as:

$\Vert \mathbf {s} \Vert ={\sqrt {s\,(s+1)}}\,\hbar$ ( $\Vert \mathbf {s} \Vert$ is the norm of the spin vector)

reference： Spin quantum number

### 泡利不相容原理

reference：Pauli exclusion principle

### 费米子

reference：Fermion

### 薛定谔的猫——Schrödinger’s cat

Schrödinger’s cat is a thought experiment, sometimes described as a paradox, devised by Austrian physicist Erwin Schrödinger in 1935.It illustrates what he saw as the problem of the Copenhagen interpretation of quantum mechanics applied to everyday objects. The scenario presents a cat that may be simultaneously both alive and dead,a state known as a quantum superposition, as a result of being linked to a random subatomic event that may or may not occur. The thought experiment is also often featured in theoretical discussions of the interpretations of quantum mechanics. Schrödinger coined the term Verschränkung (entanglement) in the course of developing the thought experiment.

reference：Schrödinger’s cat

### 德布罗意方程组

$$p={\frac {h}{\lambda }}=\hbar k={\frac {E}{c}}$$

$$E=h\nu =\hbar \omega \,!$$

$$\lambda =\frac{h}{mv} =\frac{h}{p}$$

reference：Matter wave

Wave–particle duality

De Broglie–Bohm theory

Louis de Broglie

$$F{\left( w \right) } =F\left[ f{\left( t \right) } \right] =\int_{-\infty }^{\infty } f\left( t \right) e^{-jwt} d{t}$$

$$f{\left( t \right) } =F^{-1}\left[ F{\left( w \right) } \right] =\frac{1}{2\pi } \int_{-\infty }^{\infty } F\left( w \right) e^{jwt} d{w}$$

$$L\left[ f{\left( t \right) } \right] =F{\left( s \right) } =\int_{0}^{\infty } f\left( t \right) e^{-st} d{t}$$

$$L^{-1}\left[ F{\left( s \right) } \right] =f{\left( t \right) } =\frac{1}{2\pi j } \int_{\sigma -j\infty }^{\sigma +j\infty } F\left( s \right) e^{st} d{s}$$

reference：Fourier transform

Fourier series

Joseph Fourier

Laplace transform

### 伽玛函数

$$\Gamma {\left( s \right) } =\int_{0 }^{\infty }x^{s-1} e^{-x} d{x}$$

reference：Gamma function

### 贝塔函数

$$B\left( P,Q \right) =\frac{\Gamma \left( P \right) \Gamma \left( Q \right)}{\Gamma \left( P+Q \right)} =2\int_{0}^{\frac{\pi}{2} } sin^{2P-1} xcos^{2Q-1} xd{x} (P>0，Q>0)$$

### 不完全贝塔函数

$$B(x;P,Q)=\int_{0}^{x} X^{P-1} \left( 1-X\right) ^{Q-1} dX$$

$$B(P,Q)=\int_{0}^{1} x^{P-1} \left( 1-x\right) ^{Q-1} dx$$

reference：Beta function

### 正态分布

$$f\left( x \right) =\frac{1}{\sqrt{2\pi}\sigma } e^{-\frac{\left( x-\mu \right) ^{2} }{2\sigma ^2} }$$

reference：Normal distribution

### 牛顿-莱布尼茨公式

$$\int_{a}^{b} f\left( x \right) d{x} =F{\left( b \right) } -F{\left( a \right) }$$

reference：Fundamental theorem of calculus

### 牛顿第二运动定律

$$F =ma$$

reference： Newton’s laws of motion

### 牛顿万有引力定律

$$F=\frac{GMm}{r^{2} }$$

reference：Isaac Newton

Newton’s law of universal gravitation

### 泰勒公式

$$f\left( x \right) =f\left( x_{0} \right) +{f ^{ ‘ }\left( x_{0} \right) }\left( x-x_{0} \right) +\frac{f ^{ ‘’ }\left( x_{0} \right) }{ 2 !}\left( x-x_{0} \right) ^{2}+\cdot \cdot \cdot +\frac{f ^{\left( n \right) }\left( x_{0} \right) }{ n !}\left( x-x_{0} \right) ^{n}+R_{n} \left( x \right)$$

$$R_{n} \left( x \right) =\frac{f ^{\left( n+1 \right) }\left( \xi \right) }{\left( n+1 \right) !}\left( x-x_{0} \right) ^{n+1}（\xi \in\left( a,b \right) ）$$

（高阶无穷小）时，

$$f\left( x \right) =f\left( x_{0} \right) +{f ^{ ‘ }\left( x_{0} \right) }\left( x-x_{0} \right) +\frac{f ^{ ‘’ }\left( x_{0} \right) }{ 2 !}\left( x-x_{0} \right) ^{2}+\cdot \cdot \cdot +\frac{f ^{\left( n \right) }\left( x_{0} \right) }{ n !}\left( x-x_{0} \right) ^{n}+o\left[ \left( x-x_{0} \right) ^{n} \right]$$

reference：Taylor’s theorem

### 泰勒级数

$$\sum_{n=0}^{\infty }{\frac{f ^{\left( n \right) }\left( x_{0} \right) }{ n !}\left( x-x_{0} \right) ^{n} } =f\left( x_{0} \right) +{f ^{ ‘ }\left( x_{0} \right) }\left( x-x_{0} \right) +\frac{f ^{ ‘’ }\left( x_{0} \right) }{ 2 !}\left( x-x_{0} \right) ^{2}+\cdot \cdot \cdot +\frac{f ^{\left( n \right) }\left( x_{0} \right) }{ n !}\left( x-x_{0} \right) ^{n}+\cdot \cdot \cdot$$

reference： Brook Taylor

### 齐奥尔科夫斯基公式

$${\displaystyle \Delta v=v_{\text{e}}\ln {\frac {m_{0}}{m_{f}}}}$$

$\Delta v$ is delta-v - the maximum change of velocity of the vehicle (with no external forces acting).

$m_{0}$ is the initial total mass, including propellant.

$m_{f}$ is the final total mass without propellant, also known as dry mass.

$v_{\text{e}}$ is the effective exhaust velocity.

$\ln$ refers to the natural logarithm function.

The Tsiolkovsky rocket equation, or ideal rocket equation, describes the motion of vehicles that follow the basic principle of a rocket: a device that can apply acceleration to itself (a thrust) by expelling part of its mass with high speed and thereby move due to the conservation of momentum. The equation relates the delta-v (the maximum change of velocity of the rocket if no other external forces act) with the effective exhaust velocity and the initial and final mass of a rocket (or other reaction engine).

reference：Tsiolkovsky rocket equation

Konstantin Tsiolkovsky

Non-rocket spacelaunch

Soviet space program

### 阿基米德杠杆原理

$$F_{1} L_{1} =F_{2} L_{2}$$

reference：Lever

Archimedes’ principle

Archimedes

### 玻尔兹曼公式

$$S=kln\Omega$$

reference： Boltzmann’s entropy formula

Boltzmann equation

Boltzmann constant

Boltzmann machine

Ludwig Boltzmann

reference：Boltzmann’s entropy formula

Entropy

Ludwig Boltzmann

Rudolf Clausius

Clausius–Clapeyron relation

Clausius theorem

Virial theorem

### 黎曼ζ函数

The Riemann zeta function $ζ(s)$ is a function of a complex variable $s = \sigma + it$. (The notation $s$, $\sigma$ , and t is used traditionally in the study of the $ζ$-function, following Riemann.)

The following infinite series converges for all complex numbers s with real part greater than 1, and defines $ζ(s)$ in this case:

$$\zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots \;\;\;\;\;\;\;\sigma ={\mathfrak {R}}(s)>1.!$$

It can also be defined by the integral:

$$\zeta (s)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }{\frac {x^{s-1}}{e^{x}-1}}\mathrm {d} x$$

The Riemann zeta function is defined as the analytic continuation of the function defined for $\sigma > 1$ by the sum of the preceding series.

Leonhard Euler considered the above series in 1740 for positive integer values of $s$, and later Chebyshev extended the definition to $Re(s)>1$.

The above series is a prototypical Dirichlet series that converges absolutely to an analytic function for s such that $\sigma > 1$ and diverges for all other values of s. Riemann showed that the function defined by the series on the half-plane of convergence can be continued analytically to all complex values $s \ne 1$. For $s = 1$the series is the harmonic series which diverges to $+\infty$ , and

$\lim _{s\to 1}(s-1)\zeta (s)=1.$

Thus the Riemann zeta function is a meromorphic function on the whole complex s-plane, which is holomorphic everywhere except for a simple pole at $s = 1$ with residue 1.

### 全纯函数——holomorphic function

In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain. The existence of a complex derivative in a neighborhood is a very strong condition, for it implies that any holomorphic function is actually infinitely differentiable and equal to its own Taylor series (analytic). Holomorphic functions are the central objects of study in complex analysis.

Given a complex-valued function f of a single complex variable, the derivative of f at a point z_{0} in its domain is defined by the limit:

$$f’(z_{0})=\lim {z\to z{0}}{f(z)-f(z_{0}) \over z-z_{0}}$$

This is the same as the definition of the derivative for real functions, except that all of the quantities are complex. In particular, the limit is taken as the complex number z approaches $z_{0}$ , and must have the same value for any sequence of complex values for z that approach $z_{0}$ on the complex plane. If the limit exists, we say that f is complex-differentiable at the point $z_{0}$. This concept of complex differentiability shares several properties with real differentiability: it is linear and obeys the product rule, quotient rule, and chain rule.
If f is complex differentiable at every point $z_{0}$ in an open set U, we say that f is holomorphic on U. We say that f is holomorphic at the point $z_{0}$ if it is holomorphic on some neighborhood of $z_{0}$ .We say that f is holomorphic on some non-open set A if it is holomorphic in an open set containing A.

reference： Riemann zeta function

reference： Holomorphic function

### 黎曼猜想——Riemann hypothesis

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and the complex numbers with real part 1/2. It was proposed by Bernhard Riemann (1859), after whom it is named. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields.

The Riemann hypothesis implies results about the distribution of prime numbers. Along with suitable generalizations, some mathematicians consider it the most important unresolved problem in pure mathematics (Bombieri 2000). The Riemann hypothesis, along with Goldbach’s conjecture, is part of Hilbert’s eighth problem in David Hilbert’s list of 23 unsolved problems; it is also one of the Clay Mathematics Institute’s Millennium Prize Problems.

It can also be defined by the integral:

$$\zeta (s)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }{\frac {x^{s-1}}{e^{x}-1}}\mathrm {d} x$$

The Riemann zeta function satisfies the functional equation (known as the Riemann functional equation or Riemann’s functional equation):

$$\zeta (s)=2^{s}\pi ^{s-1}\ \sin \left({\frac {\pi s}{2}}\right)\ \Gamma (1-s)\ \zeta (1-s)!$$

reference： Riemann hypothesis

reference related articles：Riemann Xi function

### Theta function

reference related articles： Theta function

### 黎曼曲面

In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together.

The main point of Riemann surfaces is that holomorphic functions may be defined between them. Riemann surfaces are nowadays considered the natural setting for studying the global behavior of these functions, especially multi-valued functions such as the square root and other algebraic functions, or the logarithm.

Every Riemann surface is a two-dimensional real analytic manifold (i.e. a surface), but it contains more structure (specifically a complex structure) which is needed for the unambiguous definition of holomorphic functions. A two-dimensional real manifold can be turned into a Riemann surface (usually in several inequivalent ways) if and only if it is orientable and metrizable. So the sphere and torus admit complex structures, but the Möbius strip, Klein bottle and projective plane do not.

There are several equivalent definitions of a Riemann surface.

1.A Riemann surface X is a complex manifold of complex dimension one. This means that X is a Hausdorff topological space endowed with an atlas: for every point x ∈ X there is a neighbourhood containing x homeomorphic to the unit disk of the complex plane. The map carrying the structure of the complex plane to the Riemann surface is called a chart. Additionally, the transition maps between two overlapping charts are required to be holomorphic.

2.A Riemann surface is an oriented manifold of (real) dimension two – a two-sided surface – together with a conformal structure. Again, manifold means that locally at any point x of X, the space is homeomorphic to a subset of the real plane. The supplement “Riemann” signifies that X is endowed with an additional structure which allows angle measurement on the manifold, namely an equivalence class of so-called Riemannian metrics. Two such metrics are considered equivalent if the angles they measure are the same. Choosing an equivalence class of metrics on X is the additional datum of the conformal structure.

1857年，他初次登台作了题为“论作为几何基础的假设”的演讲，开创了黎曼几何，并为爱因斯坦的广义相对论提供了数学基础。他在1857年升为哥廷根大学的编外教授，并在1859年狄利克雷去世后成为正教授。

reference： Riemann surface

Riemann Surfaces-PDF

### 狄利克雷函数

$$D\left( x \right) =\lim_{k \rightarrow \infty }{\left( \lim_{j \rightarrow \infty }{\left( cos\left( k!\pi x \right) \right)^{2j} } \right) }$$

$（k，j为整数）$也可以简单地表示分段函数的形式，$D(x)= 0（x是无理数）或1（x是有理数）$。

reference： Nowhere continuous function

Peter Gustav Lejeune Dirichlet

Dirichlet distribution

Dirichlet-multinomial distribution

Dirichlet process

Dirichlet series

Dirichlet convolution

### 庞加莱猜想

In mathematics, the Poincaré conjecture is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. The conjecture states:

Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

reference：Poincaré conjecture

Henri Poincaré

Poincaré disk model

Poincaré half-plane model

Poincaré group

Poincaré recurrence theorem

Poincaré duality

Poincaré inequality

Poincaré–Birkhoff–Witt theorem

《我与她的恋情》

——2016年9月28日晚

### 七桥问题

The Seven Bridges of Königsberg is a historically notable problem in mathematics. Its negative resolution by Leonhard Euler in 1736 laid the foundations of graph theory and prefigured the idea of topology.

The city of Königsberg in Prussia (now Kaliningrad, Russia) was set on both sides of the Pregel River, and included two large islands which were connected to each other and the mainland by seven bridges. The problem was to devise a walk through the city that would cross each bridge once and only once, with the provisos that: the islands could only be reached by the bridges and every bridge once accessed must be crossed to its other end. The starting and ending points of the walk need not be the same.

Euler proved that the problem has no solution. The difficulty was the development of a technique of analysis and of subsequent tests that established this assertion with mathematical rigor.

1736年29岁的欧拉向圣彼得堡科学院递交了《哥尼斯堡的七座桥》的论文，在解答问题的同时，开创了数学的一个新的分支——图论与几何拓扑，也由此展开了数学史上的新历程。七桥问题提出后，很多人对此很感兴趣，纷纷进行试验，但在相当长的时间里，始终未能解决。欧拉通过对七桥问题的研究，不仅圆满地回答了哥尼斯堡居民提出的问题，而且得到并证明了更为广泛的有关一笔画的三条结论，人们通常称之为“欧拉定理F”。

reference：Seven Bridges of Königsberg

——2016年国庆节期间

• 后记：楼主高中开始推数学公式，现在已经有不少原创数学公式和研究成果，其中泰勒级数我高中就推过，那个时候有点苗头，只可惜，上大学后明白了什么是“井底之蛙”。数学发展到今天，路几乎被数学家们踩烂了，当然，每个时代都有每个时代的命运，无论什么时代，都会有数学难题困扰着数学家，路漫漫其修远，每个人都有机会。

• 4年，一度荒废后，2016年重拾数学，经过大半年的学习，现在已经找回巅峰时期的状态，大学虽然几乎荒废了数学，但也有几个创造性的数学研究，总体来说都不是很满意。接下来的目标就是将所有数学专业的专业课程从低到高，一直研究到专业学术层，在闲暇时间的专业数学研究道路上朝着想去的方向越走越远······毫无疑问，我将不断进行数学研究，我将一直行走在这条路上，直至生命的尽头。

## 世界七大数学难题

NP完全问题、霍奇猜想、庞加莱猜想、黎曼假设、杨·米尔斯理论、纳卫尔-斯托可方程、BSD猜想。

### 3.庞加莱猜想

2006年8月，第25届国际数学家大会授予佩雷尔曼菲尔兹奖。数学界最终确认佩雷尔曼的证明解决了庞加莱猜想。

## 希尔伯特的23个数学问题——Hilbert’s problems

Hilbert’s problems are a list of twenty-three problems in mathematics published by German mathematician David Hilbert in 1900. The problems were all unsolved at the time, and several of them were very influential for 20th century mathematics. Hilbert presented ten of the problems (1, 2, 6, 7, 8, 13, 16, 19, 21 and 22) at the Paris conference of the International Congress of Mathematicians, speaking on August 8 in the Sorbonne. The complete list of 23 problems was published later, most notably in English translation in 1902 by Mary Frances Winston Newson in the Bulletin of the American Mathematical Society.

reference：Hilbert’s problems

David Hilbert

1：古希腊

2：德国

3：法国

4：美国

5：英国

6：瑞士

7：匈牙利

8：挪威

9：澳大利亚

10：苏联

11：意大利

12：印度

13：爱尔兰

14：瑞典

15：丹麦

16：捷克

17：日本

18：比利时

19：波兰

20：墨西哥

21：奥地利

22：阿拉伯

23：罗马尼亚

## 华人数学家

### 中国近现代世界著名数学家

1983年王见定教授在世界上首次提出半解析函数理论，1988年又首次提出并系统建立了共轭解析函数理论；并将这两项理论成功地应用于电场、磁场、流体力学、弹性力学等领域。此两项理论受到众多专家、学者的引用和发展，并由此引发双解析函数、复调和函数、多解析函数（k阶解析函数）、半双解析函数、半共轭解析函数以及相应的边值问题、微分方程、积分方程等一系列新的数学分支的产生，而且这种发展势头强劲有力，不可阻挡。这是中国学者对发展世界数学作出的前所未有的大范围的原创工作。

1911年10月28日生于浙江嘉兴秀水县，美籍华人，20世纪世界级的几何学家，他开创并领导着整体微分几何、纤维丛微分几何、“陈示性类”等领域的研究，在国际上享有“微分几何之父”的美誉，曾获得美国国家科学奖、“沃尔夫奖”和“邵逸夫奖”等殊荣。

1933年，匈牙利数学家乔治·塞凯赖什（George Szekeres）还只有22岁。那时，他常常和朋友们在匈牙利的首都布达佩斯讨论数学。这群人里面还有同样生于匈牙利的数学怪才——保罗·埃尔德什（PAUL ERDŐS）大神。不过当时，埃尔德什只有20岁。

## 数学软件篇

1 、数值计算软件，如matlab（商业软件），scilab(开源自由软件）等等；
2 、统计软件，如SAS（商业软件）、minitab（商业软件）、SPSS（商业软件），R（开源自由软件）等；

## 数学书籍篇

《自然哲学之数学原理》·牛顿 (Isaac Newton) , 王克迪译·北京大学出版社

《数学分析》第四版（上下册）·华东师范大学数学系·高等教育出版社

《线性代数》第六版·同济大学数学系·高等教育出版社

《复变函数》第四版·西安交通大学高等数学教研室·高等教育出版社

《概率论与数理统计》浙大第四版·高等教育出版社

《高等代数》第四版·北京大学数学系前代数小组·王萼芳·高等教育出版社

《陶哲轩实分析》·陶哲轩·人民邮电出版社

Visual Complex Analysis, by Tristan Needham

Game Theory, by Drew Fudenberg and Jean Tirole

Game Theory: Analysis of Conflict, by Roger B. Myerson

Game Theory, by Michael Maschler, Eilon Solan and Shmuel Zamir

A Course in Game Theory, by Martin Osborne and Ariel Rubinstein

《组合数学》英文第5版·Richard A.Brualdi·机械工业出版社

《微分几何》第四版·梅向明·高等教育出版社

A Course in Arithmetic, by Jean-Pierre Serre

Topology and geometry, by Glen E.Bredon

Introduction to Topological Manifolds, by John M.Lee

Topology from the Differentiable Viewpoint, by John Willard Milnor

Differential Topology, by Morris W. Hirsch

Algebraic Topology, by Allen Hatcher

Algebraic Topology, by Tammo tom Dieck

Algebraic Topology, Corr. 3rd Edition by Edwin H. Spanier

Three-Dimensional Geometry and Topology, Volume 1, William P. Thurston, Edited by Silvio Levy

Mathematical Physics ,2nd Edition by Sadri Hassani

Mathematica Methods for Physicists:A Comprehensive Guide, 7th Edition by GeorgeB.Arfken HansJ.Weber

Functional Analysis, 2nd Edition by Walter Rudin

A Course in Functional Analysis, by John B. Conway

Real and Functional Analysis, by Serge Lang

Real Analysis, Fourth Edition by H.L. Royden P.MP.M. Fitzpatrick

Real and complex analysis, by Walter Rudin

Functions of One Complex Variable, by John B.Conway

Algebra, by Serge Lang

Advanced Linear Algebra, by Steven Roman

Complex Analysis, by Lars V.Ahlfors

《简明复分析》·龚昇·中国科学技术大学出版社

Complex Analysis, by Elias M. Stein and Rami Shakarchi

《实变函数论》·周明强·北京大学出版社

《实变函数论》第五版·那汤松·高等教育出版社

《张量分析》第2版·黄克智·清华大学出版社

《近世代数》第三版·杨子胥·高等教育出版社

《解析几何》第四版·吕林根·高等教育出版社

《高等数学》·同济大学数学系·高等教育出版社

Science and Method, by Henri Poincare

《数学史通论》第二版·【美】Victor J· Katz, 李文林译·高等教育出版社

《数学史概论》·【美】伊夫斯, 李文林译·哈尔滨工业大学出版社

《古今数学思想》·【美】·克莱因·上海科学技术出版社

《数学天书中的证明》·艾格纳 (Martin Aigner) , 齐格勒 (Gunter M.Ziegler), 冯荣权& 宋春伟& 宗传明&李璐译·高等教育出版社

《费马大定理:一个困惑了世间智者358年的谜》·【英】西蒙•辛格·广西师范大学出版社

《算术探索》·【德】高斯·哈尔滨工业大学出版社

《数学指南:实用数学手册》·【德】埃伯哈德·蔡德勒·科学出版社

《数学的语言：化无形为可见》·【美】齐斯·德福林·广西师范大学出版社

《天才引导的历程:数学中的伟大定理》·【美】William Dunham·机械工业出版社

《完美的证明：一位天才和世纪数学的突破》·玛莎·葛森·北京理工大学出版社

《什么是数学:对思想和方法的基本研究》第三版·【美】R·柯朗H·罗宾, I·斯图尔特, 左平&张饴慈译·复旦大学出版社

《数学恩仇录：数学家的十大论战》·【美】哈尔·赫尔曼·复旦大学出版社

《数学大师:从芝诺到庞加莱》·埃里克•坦普尔•贝尔·上海科技教育出版社

《数学世纪:过去100年间30个重大问题》·【意】皮耶尔乔治•奥迪弗雷迪·上海科学技术出版社

## 后记

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