I enjoy doing differential equations, differential geometry and real and complex analysis. My main focus is real analysis and integration techniques.

$$ \hat{f}(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i \omega t} \, dt $$

$$ \frac{1}{A^n} = \frac{1}{(n-1)!} \int_0^\infty x^{n-1} e^{-A x} \, dx \quad \text{for } A > 0 $$

$$ \int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi}} e^{-x^2/2} \, dx = 1 $$

$$ i\hbar \frac{\partial}{\partial t} \Psi(x,t) = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} \Psi(x,t) + V(x) \Psi(x,t) $$

$$ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n $$

$$ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} $$

$$ \int_{-\infty}^{\infty} \frac{\cos x}{x^2 + 1} \, dx = \frac{\pi}{e} $$

$$ \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} $$

$$ \nabla \cdot \mathbf{B} = 0 $$

$$ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} $$

$$ \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} $$

$$ \frac{d}{dx} \left( \int_{a(x)}^{b(x)} f(x,t) \, dt \right) = f\big(x, b(x)\big) \cdot b'(x) - f\big(x, a(x)\big) \cdot a'(x) + \int_{a(x)}^{b(x)} \frac{\partial f}{\partial x}(x,t) \, dt $$

$$ \oint_\gamma f(z) \, dz = 2\pi i \sum \operatorname{Res}(f, z_k) $$

$$ e^{ix} = \cos x + i \sin x $$

$$ \int_{-\infty}^{\infty} \frac{\sin x}{x} \, dx = \pi $$

$$ \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} $$

$$ \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6} $$