width_bucket assigns a numeric value to an equiwidth histogram bucket given a range and a bucket count. It's the right tool when you need to bin continuous values into fixed-size groups — age brackets, price tiers, score ranges — without writing a chain of when expressions.

The pmod function returns the positive remainder of dividing one column by another. Unlike the standard % operator, which mirrors the sign of the dividend, pmod keeps the result non-negative whenever the divisor is positive. That makes it the right tool for hash bucketing and any cyclic indexing where a negative remainder would point you at the wrong bucket.

The hypot function computes sqrt(a² + b²) for two numeric inputs without the intermediate overflow or underflow that a naive implementation would produce. It's the standard tool for distances between points, vector magnitudes, and anywhere the Pythagorean theorem applies.

signum returns -1.0 for negative numbers, 0.0 for zero, and 1.0 for positive numbers. It's useful when you care about the direction of a value but not its magnitude — flagging gains vs. losses, classifying deltas, or branching on the sign of a difference. Spark also exposes the SQL-only aliases sign, negative, and positive for working with the sign of a column.

The factorial function returns the factorial of an integer column — the product of all positive integers up to and including the input value (n! = n × (n-1) × ... × 2 × 1). It's useful anywhere you need to compute permutations, combinations, or other counting expressions inline in a DataFrame.

The degrees and radians functions convert DataFrame columns between the two ways of measuring angles. radians turns degrees into radians; degrees does the reverse. They're the unit-conversion helpers you reach for whenever your data is in degrees but you need to feed it into Spark's trig functions, which all expect radians.

Spark Scala exposes the full set of trigonometric functions from java.lang.Math as DataFrame column functions: the basics (sin, cos, tan), their inverses (asin, acos, atan, atan2), the reciprocals (cot, csc, sec), and the hyperbolic versions of all of them. Every input and output is in radians, not degrees — use the radians function to convert if your data is in degrees.

Spark provides two exponential functions: exp computes e^x and expm1 computes e^x - 1. They're the inverses of log and log1p respectively, and you'll reach for them whenever you need to undo a log transform, compute compound growth, or work with continuous decay.

Spark provides a family of logarithm functions: log for natural log or an arbitrary base, log2 and log10 for the two most common bases, log1p for accurate results near zero, and ln (SQL-only) as an alias for the natural log. They all return Double and treat non-positive inputs as null rather than raising errors.

The sqrt, cbrt, and pow functions compute square roots, cube roots, and arbitrary powers of numeric columns. They return Double regardless of input type and behave like Java's Math.sqrt, Math.cbrt, and Math.pow — including how they handle negative inputs and special values like NaN.