first and last are aggregate functions that return the earliest or latest value in a group — but only when the data is ordered. They're most useful as window functions paired with an orderBy clause, where "first" and "last" actually mean something specific. first_value and last_value are SQL-only synonyms, and any_value returns an arbitrary value when you don't care which one you get.

min and max are aggregate functions that return the smallest and largest values in a column. min_by and max_by go a step further — they return the value of one column at the row where another column reaches its extreme, so you can answer questions like "which employee earns the highest salary?" rather than just "what is the highest salary?".

avg and mean are aggregate functions that compute the arithmetic mean of values in a numeric column. They're the workhorse functions for summarizing data — average salary by department, average order size by region, moving average of a stock price. The two functions are identical; mean is just an alias for avg.

sum is an aggregate function that totals the values in a numeric column. It's one of the most common operations in Spark Scala — used to roll up sales totals, computed metrics, running balances, and just about any group-level numeric summary.

count, countDistinct, and count_if are aggregate functions for counting rows in a Spark Scala DataFrame. count counts rows or non-null values, countDistinct counts unique values, and count_if counts rows that match a condition.

The bitwise shift functions move the bits of an integer column left or right by a fixed number of positions. They're useful for packing and unpacking flags, fast multiplication or division by powers of two, and working with binary protocol data.

e() returns Euler's number (≈ 2.71828) and pi() returns π (≈ 3.14159). They're handy when a Spark expression needs one of these mathematical constants — for circle math, exponential growth, trigonometry, and so on — without you having to hard-code the literal value.

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.