**Tetration** is a binary mathematical operator defined by the recurrence relation:

More intuitively, with *b* copies of *a*. is pronounced "*a* tetrated to *b*" or "to-the-*b* *a*."

Tetration leads to very large numbers, even with small inputs. For example, , which has 3638334640025 digits.

## Generalizing

The problem with the above definition is that works only for nonnegative integers *b*. What is , for example?

Generalizing *b* to the real numbers is a tricky and interesting problem. We present a solution proposed by Daniel Geisler of http://tetration.org/. Heavy differential calculus is ahead, so be warned. This is intended to be a gentle introduction; visit the original page if you don't need a tutorial.

Tetration is part of a class of general problems involving *function iteration*. Function iteration is defined by the recurrence relation , , so , , etc. Tetration could be defined as where . So if we can define for real , we're all set for defining continuous tetration.

First, we translate *f* so that . This simplifies a lot of the math. Then we consider the Maclaurin series (Taylor series around 0) of :

This converges to for for some radius . What we need to do is find .

### Constant term

First, because .

### First derivative

To compute , we need to find . Define for convenience:

- (Chain Rule)

Plugging in , we can take advantage of the fact that :

Since we'll see them a lot, we'll define , , etc. So we can write our latest finding as .

### Second derivative

This one is somewhat nastier. Again define :

- (Chain Rule)
- (Product Rule)
- (Product Rule)

Setting :

### Third derivative

- (Chain Rule + Product Rule)
- (Product Rule, 2x)
- (combining like terms)

## External links

- http://en.citizendium.org/wiki/Tetration
- Knuth's up-arrow notation