50 Очень Длинных Формул: Полное Решение

Хорошо, вот 50 очень длинных формул. Я постарался сделать их разнообразными, используя различные математические символы и операции. 1. \[ \int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi} \] 2. \[ \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6} \] 3. \[ \frac{d}{dx} \left( \int_{a}^{x} f(t) dt \right) = f(x) \] 4. \[ e^{i\pi} + 1 = 0 \] 5. \[ \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} \] 6. \[ \nabla \times \mathbf{B} - \frac{1}{c^2}\frac{\partial \mathbf{E}}{\partial t} = \mu_0 \mathbf{J} \] 7. \[ \oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} \] 8. \[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \] 9. \[ \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} \] 10. \[ \int_0^1 x^a (1-x)^b dx = \frac{\Gamma(a+1)\Gamma(b+1)}{\Gamma(a+b+2)} \] 11. \[ \det(A - \lambda I) = 0 \] 12. \[ \frac{1}{\Gamma(z)} = z e^{\gamma z} \prod_{n=1}^{\infty} \left( 1 + \frac{z}{n} \right) e^{-z/n} \] 13. \[ \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = \prod_{p \text{ prime}} \frac{1}{1-p^{-s}} \] 14. \[ \mathcal{F}\{f(t)\} = F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt \] 15. \[ \mathcal{L}\{f(t)\} = F(s) = \int_{0}^{\infty} f(t) e^{-st} dt \] 16. \[ \frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} \] 17. \[ \int \frac{dx}{\sqrt{a^2-x^2}} = \arcsin\left(\frac{x}{a}\right) + C \] 18. \[ \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k = (a+b)^n \] 19. \[ \frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x+h, y) - f(x, y)}{h} \] 20. \[ \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA = \oint_{\partial D} (P dx + Q dy) \] 21. \[ \nabla^2 \phi = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial \phi}{\partial r} \right) + \frac{1}{r^2 \sin\theta} \frac{\partial}{\partial \theta} \left( \sin\theta \frac{\partial \phi}{\partial \theta} \right) + \frac{1}{r^2 \sin^2\theta} \frac{\partial^2 \phi}{\partial \varphi^2} \] 22. \[ \int_0^{\infty} \frac{\sin x}{x} dx = \frac{\pi}{2} \] 23. \[ \Gamma(z+1) = z\Gamma(z) \] 24. \[ \frac{1}{2\pi i} \oint_C \frac{f(z)}{z-z_0} dz = f(z_0) \] 25. \[ \sum_{n=0}^{\infty} \frac{x^n}{n!} = e^x \] 26. \[ \int_a^b f(x) dx \approx \frac{b-a}{6} \left( f(a) + 4f\left(\frac{a+b}{2}\right) + f(b) \right) \] 27. \[ \frac{d}{dx} \left( \int_{g(x)}^{h(x)} f(t) dt \right) = f(h(x))h'(x) - f(g(x))g'(x) \] 28. \[ \text{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} dt \] 29. \[ \text{Ai}(x) = \frac{1}{\pi} \int_0^{\infty} \cos\left(\frac{t^3}{3} + xt\right) dt \] 30. \[ \text{Bi}(x) = \frac{1}{\pi} \int_0^{\infty} \left( e^{-\frac{t^3}{3} + xt} + \sin\left(\frac{t^3}{3} + xt\right) \right) dt \] 31. \[ \text{J}_{\nu}(x) = \sum_{k=0}^{\infty} \frac{(-1)^k}{k! \Gamma(k+\nu+1)} \left(\frac{x}{2}\right)^{2k+\nu} \] 32. \[ \text{Y}_{\nu}(x) = \frac{\text{J}_{\nu}(x)\cos(\nu\pi) - \text{J}_{-\nu}(x)}{\sin(\nu\pi)} \] 33. \[ \text{H}_{\nu}^{(1)}(x) = \text{J}_{\nu}(x) + i\text{Y}_{\nu}(x) \] 34. \[ \text{H}_{\nu}^{(2)}(x) = \text{J}_{\nu}(x) - i\text{Y}_{\nu}(x) \] 35. \[ \text{P}_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2-1)^n \] 36. \[ \text{L}_n(x) = \frac{e^x}{n!} \frac{d^n}{dx^n} (e^{-x} x^n) \] 37. \[ \text{H}_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} (e^{-x^2}) \] 38. \[ \text{I}_{\nu}(x) = \sum_{k=0}^{\infty} \frac{1}{k! \Gamma(k+\nu+1)} \left(\frac{x}{2}\right)^{2k+\nu} \] 39. \[ \text{K}_{\nu}(x) = \frac{\pi}{2} \frac{\text{I}_{-\nu}(x) - \text{I}_{\nu}(x)}{\sin(\nu\pi)} \] 40. \[ \text{Si}(x) = \int_0^x \frac{\sin t}{t} dt \] 41. \[ \text{Ci}(x) = -\int_x^{\infty} \frac{\cos t}{t} dt \] 42. \[ \text{Ei}(x) = \int_{-\infty}^x \frac{e^t}{t} dt \] 43. \[ \text{Li}(x) = \int_2^x \frac{dt}{\ln t} \] 44. \[ \text{FresnelC}(x) = \int_0^x \cos\left(\frac{\pi}{2}t^2\right) dt \] 45. \[ \text{FresnelS}(x) = \int_0^x \sin\left(\frac{\pi}{2}t^2\right) dt \] 46. \[ \text{DawsonF}(x) = e^{-x^2} \int_0^x e^{t^2} dt \] 47. \[ \text{LambertW}(x) e^{\text{LambertW}(x)} = x \] 48. \[ \text{Heaviside}(x) = \begin{cases} 0 & x < 0 \\ 1/2 & x = 0 \\ 1 & x > 0 \end{cases} \] 49. \[ \text{DiracDelta}(x) = \lim_{\varepsilon \to 0^+} \frac{1}{\pi} \frac{\varepsilon}{x^2+\varepsilon^2} \] 50. \[ \text{Euler-Maclaurin Formula: } \sum_{n=a}^b f(n) \approx \int_a^b f(x) dx + \frac{f(a)+f(b)}{2} + \sum_{k=1}^m \frac{B_{2k}}{(2k)!} (f^{(2k-1)}(b) - f^{(2k-1)}(a)) + R_m \]