Functions ONLY Defined by Their Integrals

1. The Gamma Function \Gamma(x)

\Gamma(x) = \int_0^\infty t^{x-1} e^{-t} \, dt, for x > 0.

Applications:

  • Generalization of factorials: \Gamma(n) = (n-1)!.
  • Used in probability distributions and statistical mechanics.
  • Found in Feynman integrals in quantum physics.

2. The Beta Function B(x, y)

B(x, y) = \int_0^1 t^{x-1} (1 - t)^{y-1} dt.

Applications:

  • Related to the Gamma function via B(x, y) = \frac{\Gamma(x) \Gamma(y)}{\Gamma(x + y)}.
  • Used in Bayesian statistics and machine learning.

3. The Error Function \operatorname{erf}(x)

\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} dt.

Applications:

  • Used in Gaussian probability distributions.
  • Appears in heat and diffusion equations.

4. The Fresnel Integrals S(x) and C(x)

S(x) = \int_0^x \sin(t^2) dt, C(x) = \int_0^x \cos(t^2) dt.

Applications:

  • Wave optics and diffraction patterns.
  • Radar signal processing.

5. The Dirichlet Integral

\int_0^\infty \frac{\sin t}{t} dt = \frac{\pi}{2}.

Applications:

  • Fourier analysis and signal processing.

6. The Bessel Functions J_n(x)

J_n(x) = \frac{1}{\pi} \int_0^\pi \cos(n t - x \sin t) dt.

Applications:

  • Solutions to differential equations in cylindrical coordinates.
  • Used in electromagnetics and fluid dynamics.

7. The Airy Function \operatorname{Ai}(x)

\operatorname{Ai}(x) = \frac{1}{\pi} \int_0^\infty \cos \left( \frac{t^3}{3} + xt \right) dt.

Applications:

  • Quantum mechanics and tunneling problems.
  • Optics and wavefront analysis.

8. The Riemann Zeta Function \zeta(s)

\zeta(s) = \frac{1}{\Gamma(s)} \int_0^\infty \frac{x^{s-1}}{e^x - 1} dx, for s > 1.

Applications:

  • Number theory and prime distribution.
  • Quantum field theory.

9. The Lambert W Function W(x)

Defined by W(x)e^{W(x)} = x, with integral representation:

W(x) = \int_0^\infty \frac{dt}{(t+1)e^{t+x}}.

Applications:

  • Used in combinatorics and graph theory.
  • Appears in quantum mechanics and delay differential equations.

10. The Polylogarithm Function \operatorname{Li}_s(x)

\operatorname{Li}_s(x) = \int_0^\infty \frac{x^t}{t^s} dt.

Applications:

  • Found in quantum field theory and statistical mechanics.
  • Used in cryptography and information theory.