Kevin Wald: The Witch of Agnesi (of Unit Height)

     
Touched by a line on top
Is a circle of unit height;
For every ray from its bottom, take
The point that's left or right
Of where ray and circle cross
And below where line meets ray.
You'll get none but the Witch of Agnesi, lad,
Stretching endlessly either way.
     
Broad though an angle drawn
From her rounded peak may be,
If it intersects her asymptote
At a pair of points Q, P,
Its size is the area
Beneath her, P to Q;
And it's none but the Witch of Agnesi, lad,
Who can make that maxim true.
     
Her height, at a distance x
From her heart, on either side,
Is one o'er the square of x plus one --
Or, if you long-divide,
The summed even powers of x,
With alternating sign;
So to measure the Witch of Agnesi, lad,
All their areas combine.
       
Now each of the areas
Of the nth-power curves that run
From naught to b is b to the n
Plus one, over n plus one;
So sum these for even n
With staggered signs to see
What's beneath the Witch of Agnesi, lad
From her heart to distance b.
       
When b's one, which corresponds
To an angle of pi o'er four,
That's one, less a third, plus a fifth, and on
For countless fractions more;
And if you dislike this sum
(Its convergence rate's quite bad)
There are lots more ways of computing pi
With the Witch of Agnesi, lad.