Once upon a time (1/t) pretty little Polynomial was strolling across a field of vectors when she came to a singularly large matrix.
Now Polly was convergent and her mother had made it an absolute condition that she must never enter such an array without her brackets on. Polly, however, who had changed her variables that morning and was feeling particularly badly behaved, ignored this condition on the grounds that it was insufficient and made her way in and amongst the complex elements.
Rows and columns enveloped her on all sides. Tangents approached her surface. She became tensor and tensor. Quite suddenly, three branches of a hyperbola touched her at a single point. She oscillated violently, lost all sense of directrix and went completely divergent. As she was reaching a turning point, she tripped over a square root which was protruding from the erf and found herself, apparently alone, in a non-Euclidian space.
She was being watched, however. That smooth operator Curly Pi, was lurking inner product. As his eyes devoured her curvilinear co-ordinates a singular expression crossed his face. "Was she convergent?" he wondered. He decided to integrate improperly at once.
Hearing a vulgar fraction behind her Polly turned around and saw Curly Pi approaching with his power series extrapolated. She could see at once, by his degenerate conic and his dissipative terms, that he was bent on no good.
"Eureka!" she gasped.
"Ho, Ho!" he said. "What a symmetric little polynomial you are. I can see you are bubbling over with secs."
"Oh, Sir" she protested, "Keep away from me. I have not got my brackets on."
"Calm yourself, my dear" said our suave operator, "Your fears are purely imaginary."
"i, i" she thought, "perhaps he is homogeneous then!"
"What order are you?" the brute demanded.
"Seventeen," replied Polly.
Curly leered. "I suppose you have never been operated on yet?"
"Of course not!" Polly cried indignantly. "I am completely convergent."
"Come, come," said Curly. "Let us go to a decimal place I know and I will take you to the limit."
"Never!" gasped Polly.
"Exchlf!" he swore using the vilest oath he knew. His patience was gone. Coshing her over the coefficient with a log until she was powerless, Curly removed her discontinuities. He stared at her significant places and began smoothing her points of inflection. Poor Polly. All was up. She felt his hand tending toward her asymptotic limit. Her convergence would soon be gone forever.
There was no mercy, for Curly was a heavyside operator. He integrated her by parts. He integrated by partial fractions. The complex beast went all the way around and did a contour integration. What an indignity! To be multiply connected on her first integration. Curly want on operating until he was absolutely and completely orthogonal.
When Polly got home that evening, her mother noticed that she was truncated in several places. But it was too late to differentiate now. As the months went by, Polly increased monotonically. Finally she generated a small but pathological function which left surds all over the place until she was driven to distraction.
The moral of our sad story is this: if you want to keep your expressions convergent, never allow them a single degree of freedom!
|The above article was reprinted from SCOPE: JOURNAL OF THE FEDERATION OF THE UNIVERSITY ASTRONOMICAL SOCIETIES. For a subscription to SCOPE write to W.H. Osborn, Box 2023, Yale Station, New Haven, Conn.|
Our thanks go to Sean Butler for providing a better attribution source.
|I have a hard copy of "Under the Double Integral" dating back to the
early 70's. The top of the article has an image of a truncated pyramid
with flames rising up before the sign of a double integral, and a line
of slaves chained together being led to and leaping into the fire.
The story is verbatim the one on your web page...