Posts

  • How to Solve Gamma Function

  • I. Theoretical Analysis Gamma Function is defined in the following, \[\Gamma(n) = \int_{0}^{\infty} \mathrm{e}^{-x}x^{n-1}\mathrm{d}x.\] Using integration by parts, one sees that: \[\Gamma(n) = -[\mathrm{e}^{-x}x^{n-1}]_0^{\infty} + (n-1)\int_{0}^{\infty} \mathrm{e}^{-x}x^{n-2}\mathrm{d}x = (n-1)\Gamma(n-1).\] From the above calculation,and note that \[\Gamma(1) = \int_{0}^{\infty} \mathrm{e}^{-x}\mathrm{d}x = 1\\ \Gamma(1/2) = \int_{0}^{\infty} \mathrm{e}^{-x}x^{-1/2}\mathrm{d}x = 2\int_{0}^{\infty} \mathrm{e}^{-y^2}\mathrm{d}y = \sqrt{\pi},\] where \(y^2 = x\). If \(n\) is a positive integer, one can obtain: \[\Gamma(n) = (n-1)(n-2)\cdots 1 \cdot \Gamma(1) = (n-1)!,\] \[\Gamma(n+\frac{1}{2}) = (n-\frac{1}{2})(n-\frac{3}{2})\cdots\frac{1}{2}\Gamma(\frac{1}{2})= (n-\frac{1}{2})(n-\frac{3}{2})\cdots\frac{1}{2}\sqrt{\pi}.\] Appendix: Integral calculus(\(y\equiv \sqrt{a}[x+(b/2a)]\)) \[\int_{-\infty}^{+\infty}\mathrm{e}^{-(ax^2+bx)}\mathrm{d}x = \int_{-\infty}^{+\infty}\mathrm{e}^{-y^2 + (b^2/4a)}\frac{1}{\sqrt{a}}\mathrm{d}y = \frac{1}{\sqrt{a}}\mathrm{e}^{b^2/4a}\int_{-\infty}^{+\infty}\mathrm{e}^{-y^2}\mathrm{d}y = \boxed{\sqrt{\frac{\pi}{a}}\mathrm{e}^{b^2/4a}}.\] If it meets a condition \(e^{-x^2} > 0\), obviously given that, \[I...
  • 100 Things You Can Do To Improve Your English

  • Don’t be afraid to make mistakes. Be confident. People can only correct your mistakes when they hear you make them. Surround yourself in English. Put yourself in an all English speaking environment where you can learn passively. The best way to learn is through speaking. Practise every day. Make yourself a study plan. Decide how much time a week you are going to spend studying and stick to it. Establish a routine. Tell your family and friends about your study plan. Get them to push you to study and also don’t let them interrupt you. Practise the 4 core skills:...
  • Canvas Clock

  • Build an analog clock using HTML canvas. Table of contents 1 Clock Intro 1.1 Create the Canvas 1.2 Code Explained 2 Clock Face 2.1 Draw a Clock Face 2.2 Code Explained 3 clock Numbers 3.1 Draw Clock Numbers 3.2 Example Explained 4 Clock Hands 4.1 Draw Clock Hands 4.2 Example Explained 5 Clock Start 5.1 Start the Clock 5.2 Example Explained 1 Clock Intro 1.1 Create the Canvas The clock needs an HTML container. Create an HTML canvas: HTML code: <!DOCTYPE html> <html> <style> canvas { -webkit-border-radius: 60px; border-radius: 60px; background-color: #FF91AF } </style> <body> <canvas id="canvas" width="400" height="400"></canvas> <script>...
  • How to install Matlab2016b on Ubuntu 16.04 LTS

  • This tutorial will explain how to install Matlab2016b on Ubuntu 16.04 LTS. The instaling files are usually divided into three packages, that is R2016b_glnxa64_dvd1.iso R2016b_glnxa64_dvd2.iso Matlab 2016b Linux64 Crack.rar Table of Contents Step 0: Before installing Step 1: Mount the directory Step 2: Start to install Step 4: Mount the second dvd file Step 5: Complete the installation Step 6: Obtain support Step 0: Before installing Unrar crack files and Read the readme.txt file. $ sudo apt install unrar-free $ mkdir crack && unrar *Crack.rar crack Step 1: Mount the directory $ sudo mkdir /media/[YOUR-USERNAME]/Matlab_R2016b $ sudo mount -t auto... Read more
  • How to configure sublime text

  • Install package Click the Tools-Command Palette menu; Input “Package Control: install Package”; Waiting until the list of package appeared; Choose your ideal package and start to install package. Configure (LaTeX) LaTeXTools Configure Markdown Markdown Editing Markdown Preview Click the Preferences-Package Settings-Mardown Editing- KeyBlings-User menu. Copy the following contents to set keys alt+m for Markdown preview. [ {"keys": ["alt+m"], "command": "markdown_preview", "args": {"target": "browser"}}, ] Click the Preferences-Package Settings-Mardown Preview- Settings-User menu. { /* Enable or not mathjax support. */ "enable_mathjax": true, /* Enable or not highlight.js support for syntax highlighting. */ "enable_highlight": true, }