50 Очень Длинных и Сложных Формул

Хорошо, вот 50 очень длинных и сложных формул, написанных в формате MathJax с использованием \( \) и \[ \]: 1. \[ \int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi} \] 2. \[ \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6} \] 3. \[ e^{i\pi} + 1 = 0 \] 4. \[ \frac{d}{dx} \int_{a(x)}^{b(x)} f(t) dt = f(b(x))b'(x) - f(a(x))a'(x) \] 5. \[ \nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0} \] 6. \[ \nabla \cdot \mathbf{B} = 0 \] 7. \[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \] 8. \[ \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \] 9. \[ \oint_C (P dx + Q dy) = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA \] 10. \[ \oint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_V (\nabla \cdot \mathbf{F}) dV \] 11. \[ \oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} \] 12. \[ \frac{\partial u}{\partial t} = \alpha \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) \] 13. \[ i\hbar \frac{\partial}{\partial t} \Psi(\mathbf{r},t) = \left( -\frac{\hbar^2}{2m} \nabla^2 + V(\mathbf{r},t) \right) \Psi(\mathbf{r},t) \] 14. \[ \det(A - \lambda I) = 0 \] 15. \[ \Gamma(z) = \int_0^\infty t^{z-1} e^{-t} dt \] 16. \[ \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} \] 17. \[ \frac{1}{\Gamma(z)} = z e^{\gamma z} \prod_{n=1}^{\infty} \left(1 + \frac{z}{n}\right) e^{-z/n} \] 18. \[ \frac{1}{2\pi i} \oint_C \frac{f(z)}{z-a} dz = f(a) \] 19. \[ \frac{1}{2\pi i} \oint_C \frac{f(z)}{(z-a)^{n+1}} dz = \frac{f^{(n)}(a)}{n!} \] 20. \[ \mathcal{F}\{f(t)\}(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt \] 21. \[ \mathcal{L}\{f(t)\}(s) = \int_0^{\infty} f(t) e^{-st} dt \] 22. \[ \frac{d^2 y}{dx^2} + P(x) \frac{dy}{dx} + Q(x) y = R(x) \] 23. \[ \mathbf{F} = m\mathbf{a} \] 24. \[ E = mc^2 \] 25. \[ \frac{\partial^2 u}{\partial t^2} = c^2 \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) \] 26. \[ \frac{1}{c^2} \frac{\partial^2 \mathbf{A}}{\partial t^2} - \nabla^2 \mathbf{A} = \mu_0 \mathbf{J} \] 27. \[ \frac{1}{c^2} \frac{\partial^2 \phi}{\partial t^2} - \nabla^2 \phi = \frac{\rho}{\epsilon_0} \] 28. \[ \left( \frac{\partial}{\partial x} + i \frac{\partial}{\partial y} \right) f(x+iy) = 0 \] 29. \[ \frac{1}{2\pi} \int_{-\pi}^{\pi} |f(x)|^2 dx = \sum_{n=-\infty}^{\infty} |c_n|^2 \] 30. \[ \text{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} dt \] 31. \[ \text{erfc}(x) = 1 - \text{erf}(x) = \frac{2}{\sqrt{\pi}} \int_x^\infty e^{-t^2} dt \] 32. \[ \text{Ai}(x) = \frac{1}{\pi} \int_0^\infty \cos\left(\frac{t^3}{3} + xt\right) dt \] 33. \[ \text{Bi}(x) = \frac{1}{\pi} \int_0^\infty \left[ e^{-\frac{t^3}{3} + xt} + \sin\left(\frac{t^3}{3} + xt\right) \right] dt \] 34. \[ J_\nu(x) = \sum_{k=0}^\infty \frac{(-1)^k}{k! \Gamma(k+\nu+1)} \left(\frac{x}{2}\right)^{2k+\nu} \] 35. \[ Y_\nu(x) = \frac{J_\nu(x) \cos(\nu\pi) - J_{-\nu}(x)}{\sin(\nu\pi)} \] 36. \[ H_\nu^{(1)}(x) = J_\nu(x) + i Y_\nu(x) \] 37. \[ H_\nu^{(2)}(x) = J_\nu(x) - i Y_\nu(x) \] 38. \[ P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2-1)^n \] 39. \[ L_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (e^{-x} x^n) \] 40. \[ H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} \] 41. \[ \text{Li}_s(z) = \sum_{k=1}^\infty \frac{z^k}{k^s} \] 42. \[ \text{Ei}(x) = \int_{-\infty}^x \frac{e^t}{t} dt \] 43. \[ \text{Si}(x) = \int_0^x \frac{\sin t}{t} dt \] 44. \[ \text{Ci}(x) = -\int_x^\infty \frac{\cos t}{t} dt \] 45. \[ \text{FresnelC}(x) = \int_0^x \cos\left(\frac{\pi}{2} t^2\right) dt \] 46. \[ \text{FresnelS}(x) = \int_0^x \sin\left(\frac{\pi}{2} t^2\right) dt \] 47. \[ \text{LambertW}(z) = w \text{ such that } z = w e^w \] 48. \[ \text{hypergeom}(a_1, \dots, a_p; b_1, \dots, b_q; z) = \sum_{n=0}^\infty \frac{(a_1)_n \dots (a_p)_n}{(b_1)_n \dots (b_q)_n} \frac{z^n}{n!} \] 49. \[ \text{MeijerG}_{p,q}^{m,n} \left( \begin{matrix} a_1, \dots, a_p \\ b_1, \dots, b_q \end{matrix} \middle| z \right) = \frac{1}{2\pi i} \int_L \frac{\prod_{j=1}^m \Gamma(b_j-s) \prod_{j=1}^n \Gamma(1-a_j+s)}{\prod_{j=m+1}^q \Gamma(1-b_j+s) \prod_{j=n+1}^p \Gamma(a_j-s)} z^s ds \] 50. \[ \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right) \]