1

I just came across the following on page 176 of "An Introduction to Partial Differential Equations" by Yehuda Pinchover and Jacob Rubinstein:

Page 176

Here I can not understand the notation for the boundary conditions (7.10) and the harmonic function afterwards. Precisely, I do not understand the meaning of the variable $n$ on the left side of the equations (as an upper index). Also, in the second condition of (7.10), I do not understand the meaning of $y$ on the left side of the equation (as a bottom index).

I have a thought $n$ should mean $n$th-derivative and $y$ shows the variable in respect to which the differentiation is done. (based on other notations, I have seen) But I feel that here the meaning should be different.

It could be great if someone could spell the whole equations with words, so there is no ambiguity.

Tried to search for similar notation for this context throughout the book and in other places on the web, but with no success.

Thank you!

Dando18
  • 5,368
victor175
  • 125

1 Answers1

1

I'm interpreting that as the superscript $n$ just being an index, i.e., a way of indicating that for each $n$ there is a different $u$, i.e., $u$ depends on $n$ just as $u$ depends on $x$ and $y$.

It's perhaps more custom to use $u_n$, but the subscript area was already reserved for the partial derivative notation in this context.

Another notation I've seen in similar contexts is $u(x,y;n)$. For example: $$u(x,y;n) = \frac1{n^2}\sin nx \sinh ny$$

  • If n was an index, this would suggest that for different n, the function is different, so it is just another variable.. I guess it would be familiar to see this in a sum, or something like this, but the way it is here, it seems a bit strange.. Could you somehow explain further, so this ambiguity is gone? – victor175 Jun 22 '17 at 19:17
  • @victor175, see here for more details. –  Jun 22 '17 at 19:29
  • What I was confused about is, that indices do not stand out in the equations just like that, so I did not realize what is happening. After reading your answer and reading more, I realized that this actually was a sum, and this cleared it all out. – victor175 Jul 11 '17 at 06:09