Mathematics

Calculus

\(\int udv = u\cdot v-\int du\cdot v\)

\(\displaystyle\Gamma(n)=\int_{0}^{\infty}x^{n-1}e^{-x}dx=(n-1)!\)

  • \(cos ^2(i\theta)+sin^2(i\theta) = cosh^2(\theta)- sinh^2(\theta)\)

    \(cos(i\theta) = cosh(\theta)\)

    \(sin(i\theta) = i\cdot sinh(\theta)\)

    \(sin(x) = \dfrac{e^{jx}-e^{-jx}}{2j}\)

    \(sinh(x) = \dfrac{e^{x}-e^{-x}}{2}\)

\(2\cos \alpha \cos \beta= \cos(\alpha-\beta)+\cos(\alpha+\beta)\)

\(2\sin \alpha \cos \beta= \sin(\alpha+\beta)+\sin(\alpha-\beta)\)

\(2\sin \alpha \sin \beta = \cos(\alpha-\beta)-\cos(\alpha+\beta)\)

  • \(\dfrac{sin(x)}{x}\)

    zero-crossovers : \(x=K \pi\), \(K \in I - \{0\}\)

  • \(e^{-1}\)

    =0.3678

Complex Analysis

\(e^{j2\pi k}e^{j\theta} = e^{j\theta}\)

\(i^n+i^{n+1}+i^{n+2}+i^{n+3}=0\)

  • root of unity

    sum of ‘n’ roots of unity = 0

    product of ‘n’ roots of unity = \((-1)^{n-1}\)

  • CR equation

    \(f'(z_o) = u_x +i v_x=v_y-iu_y=e^{-i\theta}(u_r + iv_r)\)

    • cartesian

      \(u_x=v_y \\ v_x = -u_y\)

    • polar

      \(u_r=\frac{1}{r}v_{\theta} \\ v_r=-\frac{1}{r}u_{\theta}\)

  • milne thompson method

  • residue theorem

    singular point → point where complex function becomes non-analytic

    laurent series around singularity: $\( f(z) = \sum \limits _{n=0}^{\infty} a_{n} (z-a)^{n} +\sum \limits _{m=1}^{\infty} b_{-m} (z-a)^{-m} \)$

    \[\displaystyle \int_C f (z) dz = 2 \pi i \cdot\text{(sum of residues at the singular points within $z_o$ )}\]
    \[\text{residue}(f(z_o))=\text{coefficient of $(z-a)^{-1}$}=\lim \limits_{z\rightarrow z_o} \dfrac{1}{(n-1)!}\Big[ \dfrac{d^{n-1}}{dz^{n-1}}[(z-z_o)^nf(z)]\Big]\]
ODE

order → highest order derivative

degree → degree of highest order differential coefficient provided equation is polynomial in all differential coefficients

  • integration and differentiation

    \(\displaystyle \int \dfrac{1}{\sqrt{x^2+a^2}} dx = \sin^{-1} \Big(\frac{x}{a}\Big) + C\)

    \(\displaystyle \int \dfrac{1}{\sqrt{x^2-a^2}} dx = \cos^{-1} \Big(\frac{x}{a}\Big) + C\)

    \(\displaystyle \int \dfrac{1}{x^2-a^2} dx = \dfrac{1}{a}\tan^{-1} \Big(\frac{x}{a}\Big) + C\)

  • variable separable

    \(f(x) dx +g(y)dy =0\)

  • exact differential equation

    \({M\left( {x,y} \right)dx + N\left( {x,y} \right)dy }= 0\)

    \(\dfrac{{\partial M}}{{\partial x}} = \dfrac{{\partial N}}{{\partial y}}\)

    \(\displaystyle\int dv = \int_\text{y = const} Mdx + \int_\text{\text{independent of x}} Ndy\)

  • \(\dfrac{dy}{dx}+p(x)y=r(x)\)

    \(\displaystyle y e^{-\int p(x) dx } = \Big[\int r(x)e^{\int p(x) dx } dx +C \Big]\)

  • LDE of higher order

    \(y=CF+PI\)

    • CF

      CF is solution of \((D^n +a_1D^{n-1}+\dots +k_nD^0)y=0\)

      if roots are real and distinct : \(y =c_1e^{m_1x}+c_2e^{m_2x}\)

      if roots are real and equal : \(y =(c_1+c_2x)e^{mx}\)

      if roots are complex conjugate: \(y =(c_1\cos(\beta x)+c_2sin(\beta x))e^{\alpha x}\)

    • PI

      PI is solution of \((D^n +a_1D^{n-1}+\dots +k_nD^0)y=x\)

Linear Algebra

  • rank

    atleast one minor of A of order r doesn’t vanish → rank(A) = r

  • eigen vectors

    \([A] [X] =\lambda [X]\)

Probability

  • binomial distribution

    \(\mu = np\)

    \(\sigma ^2 = npq\)

    \(\displaystyle P(r)= \binom{n}{r} p^r q^{n-r}\)