🧮 Favorite and important mathematical formulas

**Euler's identity a'la the world's most beautiful mathematical formula:**

$$ {\displaystyle e^{i\pi }+1=0} $$

**Equivalence of mass and energy described by Albert Einstein:**

$$ E=mc^{2} $$

**Fourier transform:**

$$ {\displaystyle {\hat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)\ e^{-i2\pi \xi x}\,dx,\quad \forall \ \xi \in \mathbb {R} .} $$

The positional-spatial Schrödinger equation for a single non-relativistic particle in one dimension:

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

Residue (complex analysis):

$$ {\displaystyle \mathrm {Res} (f,z_{0})={\frac {1}{2\pi i}}\oint \limits _{\gamma }f(z)\ \operatorname {d} z,} $$

Maxwell's equations:

                                                                                Gauss's law

$$ \oiint _{\partial \Omega }\mathbf {E} \cdot \mathrm {d} \mathbf {S} ={\frac {1}{\varepsilon _{0}}}\iiint _{\Omega }\rho \,\mathrm {d} V $$

                                                                 Gauss's law for magnetism

$$ \oiint _{\partial \Omega }\mathbf {B} \cdot \mathrm {d} \mathbf {S} =0 $$

                                      Maxwell-Faraday equation (Faraday's Law of Induction)

$$ \oint _{\partial \Sigma }\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}=-{\frac {\mathrm {d} }{\mathrm {d} t}}\iint _{\Sigma }\mathbf {B} \cdot \mathrm {d} \mathbf {S} $$

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