Special functions

• What Are the Main Properties of the Digamma Function?
  The digamma function is the logarithmic derivative of the gamma function. Its recurrence, reflection, multiplication, and series formulas help evaluate it.
• How to Prove the Beta–Gamma Function Identity
  For positive x and y, transform the product of two gamma integrals using s = u + v and t = u/(u + v) to derive the beta–gamma identity.
• How to Evaluate a Limit Involving a Hypergeometric Function
  A hypergeometric limit depends on its parameters and approach point. Learn when continuity applies and how Gauss's summation formula evaluates limits at 1.
• What Is the Normal CDF Φ Called?
  The normal CDF Φ is usually called the standard normal cumulative distribution function; it is related to erf, while Φ⁻¹ is the probit.
• How to Integrate cos(1/x)
  The antiderivative of cos(1/x) is x cos(1/x) + Si(1/x) + C, where Si is the sine integral special function.
• How to Integrate a Physicists’ Hermite Polynomial with e^(-x²)
  For physicists’ Hermite polynomials, ∫Hₙ(x)e^(−x²)dx = −e^(−x²)Hₙ₋₁(x)+C for n≥1; n=0 uses the error function.
• How to Calculate the Factorial of a Noninteger
  The factorial of a noninteger is calculated with the Gamma function, giving values such as 1/2! = √π/2 and 3.5! = 105√π/16.
• How to Find the Support of a Function
  Find a function’s support by taking the closure of the points where it is nonzero; learn how endpoints, domains, and almost-everywhere definitions affect it.
• Expressing Sine and Cosine as Integrals of Bessel Functions
  For x ≥ 0, sine and cosine have exact integral formulas using Bessel functions of orders −1/2 and 1/2, respectively.
• How to Expand the Lower Incomplete Gamma Function for s < 0
  For negative noninteger s, the lower incomplete gamma function has a convergent series in powers of x; its defining integral from zero diverges.
• How to Differentiate Bessel Functions
  Differentiate Bessel functions using recurrence identities, with formulas for J, Y, I, and K and a worked example involving a weighted function.
• How to Derive Bessel Function Recurrences from the Generating Function
  Differentiate the Laurent generating function for J_n(x), match coefficients, and obtain the standard order and derivative recurrences.
• How to Derive the Hermite Polynomial Derivative Relation
  Differentiate the physicists’ Hermite equation to show H_n' solves the degree-(n−1) equation, then compare leading coefficients to get H_n'=2nH_{n−1}.