Airy Function Derivative Calculator Ai'(x), Bi'(x)

Enter a real number x to compute the first derivatives of the Airy functions, Ai'(x) and Bi'(x), via power series. Reference values of Ai(x) and Bi(x) and a graph of Ai'(x) are also shown.

Input

Enter a real number x to compute the first derivatives of the Airy functions, Ai''(x) and Bi''(x).

Value of x

Enter a real number (very large magnitudes are rejected for accuracy).

Result

Ai'(x) at x = -1

-0.0101605671

Bi'(-1)

0.5923756264

Ai(-1)

0.5355608833

Bi(-1)

0.1039973895

Curve of Ai''(x)

How it works

The Airy functions Ai(x) and Bi(x) are the two independent solutions of the differential equation y'' = x y. Ai'(x) and Bi'(x) denote their first derivatives.
Both Ai(x) and Bi(x) are entire functions (analytic for all real x), so they can be evaluated with Maclaurin series of infinite radius of convergence. This tool uses the two standard power series f(x) and g(x) and differentiates them term by term to obtain Ai'(x) and Bi'(x).
Constants matching the known origin values Ai(0) and Ai'(0) are used as coefficients. For positive x, Ai'(x) decays rapidly toward 0 while Bi'(x) grows rapidly; for negative x, both oscillate.
The series loses accuracy for large magnitudes of x due to cancellation of terms, so the allowed range of x is limited. Values outside this range return an error.

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