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\[

T(x, y) = \sum_{n = 1}^{\infty}c_n\exp\left(-\frac{\pi n}{\ell} y\right)

\sin\left(\frac{\pi n}{\ell}x\right)

\]

How does one obtain the results of finite plate by making the change of variables \(d - y\) for \(y\) and considering the linit as \(d\to\infty\)?

Making that sub we have \(\exp(-\lambda_nd)\exp(\lambda_ny)\). If we take the limit as d goes to inifinity, we get 0. Therefore, \(T(x,y) = 0\). This doesn't seem correct.