## Introduction

In mathematics, the **exponential function** is the function *e*^{x}, where *e* is the number (approximately 2.718281828) such that the function *e*^{x} is its own derivative. The exponential function can be characterized in many ways, one of the most common characterizations is with the infinite Taylor series.

A **Taylor series** is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.

The concept of a Taylor series was formally introduced by the English mathematician Brook Taylor in 1715. If the Taylor series is centered at zero, then that series is also called a **Maclaurin series**, named after the Scottish mathematician Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18^{th} century.

I implement the exponential function in two different ways, one with recursion and the other without, I used Common Table Expression; for more details about using and implementing CTE check this nice article: CTE in SQL Servers.

## Implement with CTE

To find out what I do in my code, as the picture shows I need to calculate the factorial of digits and power of them, I create a table with an identity row by cte:

As default number of recursion in cte is 100, I limit the identity number to 100, so the series continues 100 times.

;WITH CTE AS
(SELECT cast(1.0 as float) AS rowNumber
UNION ALL
SELECT cast(cast(rowNumber as float)+1 as float)
from CTE
WHERE rowNumber < 100)
select *
FROM CTE

And the result is numbers 1 to 100:

Then for each number I need its factorial. For this I add an additional column, and it is the result of multiplication with the last rows for each row:

;WITH CTE AS
(SELECT cast(1.0 as float) AS i,
cast(1.0 as float) AS factorial
UNION ALL
SELECT cast(cast(i as float)+1 as float),
cast(cast(factorial as float)*(cast(i as float)+1)as float)
FROM CTE
WHERE i < 100)
select * from CTE

And the result is as below:

At last for each row I want the power of `rowNumber`

for x and I use the power function.

As first the row which is created is the result of @x ^1. In the second row I start the power with rowNumber +1.

declare @x float = 3
;WITH CTE AS
(SELECT cast(1.0 as float) AS rowNumber,
cast(1.0 as float) AS factorial,
cast(@x as float)as [power]
UNION ALL
SELECT cast(cast(rowNumber as float)+1 as float),
cast(cast(factorial as float)*(cast(rowNumber as float)+1)as float),
cast(power(cast(@x as float),cast(rowNumber+1 as float)) as float)
FROM CTE
WHERE rowNumber < 100)
select * from CTE

And the result is as below:

At last I added the division of power to the factorial in each row with others by dynamic concatenation:

create FUNCTION dbo.EpowerX(@x int)
RETURNS FLOAT
AS
BEGIN
declare @Result float
;WITH N AS
(SELECT cast(1.0 as float) AS i,
cast(1.0 as float) AS f,
cast(@x as float)as g
UNION ALL
SELECT cast(cast(i as float)+1 as float),
cast(cast(f as float)*(cast(i as float)+1)as float),
cast(power(cast(@x as float),cast(i+1 as float)) as float)
FROM N
WHERE i < 100)
select @Result=isnull(@Result,cast(@x as float))+ cast(g as float)/cast(f as float)
from N
where i !=1
OPTION (MAXRECURSION 0);
RETURN @Result + 1
END
select cast(cast(dbo.EpowerX(30) as float) as decimal(38,3)) as myex
select cast(cast(EXP(30)as float) as decimal(38,3)) sqlexp

And the result is :

I hope it was a useful article and you enjoyed the T-SQL programming.

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