Basis.Functions.tex

\begin{equation}
\begin{split}
  \hat{\varphi}_1(\hat{x},\hat{y}) &= 1-10\hat{x}^3-10\hat{y}^3+15\hat{x}^4-30\hat{x}^2\hat{y}^2+15\hat{y}^4-6\hat{x}^5+30\hat{x}^3\hat{y}^2+30\hat{x}^2\hat{y}^3-6\hat{y}^5 \\
  \hat{\varphi}_2(\hat{x},\hat{y}) &= 10\hat{x}^3-15\hat{x}^4+15\hat{x}^2\hat{y}^2+6\hat{x}^5-15\hat{x}^3\hat{y}^2-15\hat{x}^2\hat{y}^3 \\
  \hat{\varphi}_3(\hat{x},\hat{y}) &= 10\hat{y}^3+15\hat{x}^2\hat{y}^2-15\hat{y}^4-15\hat{x}^3\hat{y}^2-15\hat{x}^2\hat{y}^3+6\hat{y}^2 \\
  \hat{\varphi}_4(\hat{x},\hat{y}) &= \hat{x}-6\hat{x}^3-11\hat{x}\hat{y}^2+8\hat{x}^4+10\hat{x}^2\hat{y}^2+18\hat{x}\hat{y}^3-3\hat{x}^5+\hat{x}^3\hat{y}^2-10\hat{x}^2\hat{y}^3-8\hat{x}\hat{y}^4 \\
  \hat{\varphi}_5(\hat{x},\hat{y}) &= \hat{y}-11\hat{x}^2\hat{y}-6\hat{y}^3+18\hat{x}^3\hat{y}+10\hat{x}^2\hat{y}^2+8\hat{y}^4-8\hat{x}^4\hat{y}-10\hat{x}^3\hat{y}^2+\hat{x}^2\hat{y}^3-3\hat{y}^5 \\
  \hat{\varphi}_6(\hat{x},\hat{y}) &= -4\hat{x}^3+7\hat{x}^4-\frac{7}{2}\hat{x}^2\hat{y}^2-3\hat{x}^5+\frac{7}{2}\hat{x}^3\hat{y}^2+\frac{7}{2}\hat{x}^2\hat{y}^3 \\
  \hat{\varphi}_7(\hat{x},\hat{y}) &= -5\hat{x}^2\hat{y}+14\hat{x}^3\hat{y}+\frac{37}{2}\hat{x}^2\hat{y}^2-8\hat{x}^4\hat{y}-\frac{37}{2}\hat{x}^3\hat{y}^2-\frac{27}{2}\hat{x}^2\hat{y}^3 \\
  \hat{\varphi}_8(\hat{x},\hat{y}) &= -5\hat{x}\hat{y}^2+\frac{37}{2}\hat{x}^2\hat{y}^2+14\hat{x}\hat{y}^3-\frac{27}{2}\hat{x}^3\hat{y}^2-\frac{37}{2}\hat{x}^3\hat{y}^2-8\hat{x}\hat{y}^4 \\
  \hat{\varphi}_9(\hat{x},\hat{y}) &= -4\hat{y}^3-\frac{7}{2}\hat{x}^3+7\hat{y}^4+\frac{7}{2}\hat{x}^3\hat{y}^2+\frac{7}{2}\hat{x}^2\hat{y}^3-3\hat{y}^5 \\
  \hat{\varphi}_{10}(\hat{x},\hat{y}) &= \frac{1}{2}\hat{x}^2-\frac{3}{2}\hat{x}^3+\frac{3}{2}\hat{x}^4-\frac{3}{2}\hat{x}^2\hat{y}^2-\frac{1}{2}\hat{x}^5+\frac{3}{2}\hat{x}^3\hat{y}^2+\hat{x}^2\hat{y}^3 \\
  \hat{\varphi}_{11}(\hat{x},\hat{y}) &= \hat{x}\hat{y}-4\hat{x}^2\hat{y}-4\hat{x}\hat{y}^2+5\hat{x}^3\hat{y}+10\hat{x}^2\hat{y}^2+5\hat{x}\hat{y}^3-2\hat{x}^4\hat{y}-6\hat{x}^3\hat{y}^2-6\hat{x}^2\hat{y}^3-2\hat{x}\hat{y}^4 \\
  \hat{\varphi}_{12}(\hat{x},\hat{y}) &= \frac{1}{2}\hat{y}^2-\frac{3}{2}\hat{y}^3-\frac{3}{2}\hat{x}^2\hat{y}^2+\frac{3}{2}\hat{y}^4+\hat{x}^3\hat{y}^2+\frac{3}{2}\hat{x}^2\hat{y}^3-\frac{1}{2}\hat{y}^5 \\
  \hat{\varphi}_{13}(\hat{x},\hat{y}) &= \frac{1}{2}\hat{x}^3-\hat{x}^4+\frac{1}{4}\hat{x}^2\hat{y}^2+\frac{1}{2}\hat{x}^5-\frac{1}{4}\hat{x}^3\hat{y}^2-\frac{1}{4}\hat{x}^2\hat{y}^3 \\
  \hat{\varphi}_{14}(\hat{x},\hat{y}) &= \hat{x}^2\hat{y}-3\hat{x}^3\hat{y}-\frac{7}{2}\hat{x}^2\hat{y}^2+2\hat{x}^4\hat{y}+\frac{7}{2}\hat{x}^3\hat{y}^2+\frac{5}{2}\hat{x}^2\hat{y}^3 \\
  \hat{\varphi}_{15}(\hat{x},\hat{y}) &= \frac{5}{4}\hat{x}^2\hat{y}^2-\frac{3}{4}\hat{x}^3\hat{y}^2-\frac{5}{4}\hat{x}^2\hat{y}^3 \\
  \hat{\varphi}_{16}(\hat{x},\hat{y}) &= \frac{5}{4}\hat{x}^2\hat{y}^2-\frac{5}{4}\hat{x}^3\hat{y}^2-\frac{3}{4}\hat{x}^2\hat{y}^3 \\
  \hat{\varphi}_{17}(\hat{x},\hat{y}) &= \hat{x}\hat{y}^2-\frac{7}{2}\hat{x}^2\hat{y}^2-3\hat{x}\hat{y}^3+\frac{5}{2}\hat{x}^3\hat{y}^2+\frac{7}{2}\hat{x}^2\hat{y}^3+2\hat{x}\hat{y}^4 \\
  \hat{\varphi}_{18}(\hat{x},\hat{y}) &= \frac{1}{2}\hat{y}^3+\frac{1}{4}\hat{x}^2\hat{y}^2-\hat{y}^4-\frac{1}{4}\hat{x}^3\hat{y}^2-\frac{1}{4}\hat{x}^2\hat{y}^3+\frac{1}{2}\hat{y}^5 \\
  \hat{\varphi}_{19}(\hat{x},\hat{y}) &= 16\hat{x}^2\hat{y}-32\hat{x}^3\hat{y}-32\hat{x}^2\hat{y}^2+16\hat{x}^4\hat{y}+32\hat{x}^3\hat{y}^2+16\hat{x}^2\hat{y}^3 \\
  \hat{\varphi}_{20}(\hat{x},\hat{y}) &= -16\hat{x}\hat{y}^2+32\hat{x}^2\hat{y}^2+32\hat{x}\hat{y}^3-16\hat{x}^3\hat{y}^2-32\hat{x}^2\hat{y}^3-16\hat{x}\hat{y}^4 \\
  \hat{\varphi}_{21}(\hat{x},\hat{y}) &= \sqrt{2}\left(8\hat{x}^2\hat{y}^2-8\hat{x}^3\hat{y}^2-8\hat{x}^2\hat{y}^3\right).
\end{split}
\label{eqn:Argyris}
\end{equation}