Special Function Calculators

Professional online calculators for advanced mathematical special functions

Gamma Functions

Γ
Gamma Function (Γ(x)) Popular
Generalization of factorial for real and complex numbers
1/Γ
Inverse Gamma Function
Reciprocal Gamma function 1/Γ(x) for numerical stability
ln Γ
Log Gamma Function (ln Γ(x))
Logarithmic Gamma function to prevent overflow issues
ψ
Digamma Function (ψ(x))
First derivative of the logarithmic Gamma function
ψ'
Trigamma Function (ψ'(x))
Second derivative of the logarithmic Gamma function

Bessel Functions

J
Bessel J Function (Jᵥ(z)) Popular
Bessel function of the first kind - solutions to cylindrical problems
Y
Bessel Y Function (Yᵥ(z))
Bessel function of the second kind (Neumann function)
I
Modified Bessel I (Iᵥ(z))
Modified Bessel function of the first kind
K
Modified Bessel K (Kₙ(x))
Modified Bessel function of the second kind
j
Spherical Bessel J (jᵥ(x))
Spherical Bessel functions of the first kind
y
Spherical Bessel Y (yᵥ(x))
Spherical Bessel functions of the second kind
Ie
Scaled Bessel Ie
Exponentially scaled modified Bessel function
Ke
Scaled Bessel Ke
Exponentially scaled modified Bessel function
Je
Scaled Bessel Je
Exponentially scaled Bessel function
Ye
Scaled Bessel Ye
Exponentially scaled Bessel function
H
Hankel Function
Hankel functions - linear combinations of Bessel functions

Zeta Functions

ζ
Riemann Zeta Function (ζ(s)) Popular
Key to prime number distribution and number theory

Airy Functions

Ai
Airy Function (Ai(x), Bi(x))
Solutions to the Airy differential equation
Ai'
Derivative Airy Function (Ai'(x), Bi'(x))
Derivatives of the Airy functions

Activation Functions

σ
Sigmoid Function (σ(x)) Popular
Classic activation function for neural networks
Logit Function
Inverse function of the sigmoid function
Softsign Function
Alternative to the tanh activation function
Softmax Function
Normalization for multi-class classification
σ'
Derivative Sigmoid
Gradient of the sigmoid function for backpropagation
Derivative Softsign
Gradient of the softsign function

Fibonacci Functions

F
Fibonacci Function (F(n)) Popular
Calculate the classic Fibonacci number sequence
Fibonacci Table
Generate comprehensive Fibonacci number tables

Struve Functions

H
Struve Function (Hᵥ(x))
Special functions related to Bessel functions
L
Modified Struve Function (Lᵥ(x))
Modified variant of the Struve functions
Struve Function Table
Generate Struve function value tables
Modified Struve Table
Generate modified Struve function value tables

About Special Functions

Special functions are mathematical functions of fundamental importance in various fields of science and engineering. They appear as solutions to differential equations:

  • Physics - Wave equations
  • Engineering - Vibration analysis
  • Statistics - Probability distributions
  • Quantum Mechanics - Schrödinger equation
  • AI/ML - Activation functions
  • Number Theory - Analytic functions
Important Function Classes
Gamma Functions
Γ(n) = (n-1)!
Generalization of factorial
Bessel Functions
Jᵥ(x), Yᵥ(x), Iᵥ(x), Kᵥ(x)
Cylindrical symmetry problems
Zeta Functions
ζ(s) = Σ n⁻ˢ
Prime number distribution
Activation Functions
σ(x) = 1/(1+e⁻ˣ)
Neural networks
Tip: Special functions often have special symmetries and recurrence relations that can be exploited for numerical calculations.
Quick Reference
Γ(n) = (n-1)!
Gamma
J₀(x)
Bessel
ζ(2) = π²/6
Zeta
σ(x)
Sigmoid
Ai(x), Bi(x)
Airy
Related Calculator Categories
Mathematical Functions
Advanced Mathematics
Applications