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\title{Mathematical Equations and Expressions}
\author{Avinash More} 

\maketitle

PI: $\pi$ \\
Area of a circle: $$ A = \pi r^2 $$ \\
\textbf{Trigonometric functions:} \\
\\
Sin: $$y = \sin{x} $$
Cosine: $$y = \cos{x} $$
Tan: $$y = \tan{x} $$
\textbf{Log functions}:\\
\\
Log: $$ \log{x} $$ 
Natural Log: $$ \ln{x} $$

\textbf{Roots:}\\
\\
Square root: $$ \sqrt{x} $$
Cube root: $$\sqrt[3]{x} $$
Nested root: $$\sqrt{1 + \sqrt{x}}$$

\textbf {Fractions:} \\
\\ 
$$\frac{2}{3} $$ 
$$\frac{\sqrt[3]{x+1}}{\sqrt[4]{x-1}}$$
$$\sqrt{\frac{x}{x^2 + 2x +1}}$$
\\

\textbf{Equations That Changed the World} \\
\\

\textbf{ 1: Pythagorean Theorem: }\\
If a and b are non-hypotenuse sides of a right angle triangle and c is a hypotenuse of the same triangle then, 
$$ c^2 = a^2 + b^2 $$

\textbf{2: The logarithm and its identities: }\\
$$\log{xy} = \log{x} + \log {y}$$

\textbf{3: The fundamental theorem of calculus:}\\
$$\frac{df}{dt} = \lim_{h\to 0} {\frac{f(t+h) - f(t)}{h}}$$

\textbf{4: Newton's universal law of gravitation:} \\ 
$$ F = G \frac{m_1m_2}{d^2}$$

\textbf{5: The origin of complex numbers:}\\
$$ i^2 = -1$$

\textbf{6: The normal distribution:} \\
$$\Phi(x) = \frac{1}{\sqrt{2\pi\sigma}}e^{\frac{{(x-\mu)}^2}{2\sigma^2}}$$

\textbf{7: The wave equation:} \\
$$\frac{\delta^2u}{\delta t^2} = c^2 \frac{\delta^2u}{\delta x^2}$$

\textbf{8: The Fourier transform:}\\
$$f(\zeta) = \int\limits_{-\infty}^{\infty}f(x)e^{-2\pi i x \zeta} dx $$

\textbf{9: Einstein's theory of relativity:}\\
$$E = mc^2$$

\textbf{10. Summation: }$$\sum_{n=1}^{\infty} 2^{-n} = 1$$ \\

\textbf{11. Matrix:}\\
\[
M=
  \begin{bmatrix}
    1 & 2 & 3 & 4 & 5 \\
    3 & 4 & 5 & 6 & 7
  \end{bmatrix}
\]

\textbf{12. Recursive:} \\ 
\[
n = \cfrac{1}{\frac{2n-1}{n} +
         \cfrac{1}{
         \begin{array}{ccc}
           \frac{2n-3}{n} & & \\
           & \ddots & \\
           & & \cfrac{1}{\frac{2n-n}{n} }
         \end{array}
       }}
\]

\textbf{Integration: }$$\int\limits_0^1 x^2 + y^2 \ dx $$ \\



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