Would any one tell me what is the $\partial^2U/\partial X^2$ where $U(X,Z)=\frac{1}{W(Z)}\psi\left(\frac{X-X_c(Z)}{W(Z)},\xi(Z)\right)e^{i\phi(X,Z)}$
What is the derivative of this function?
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calculus
ordinary-differential-equations
1 Answers
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Since $U(X,W)=R(X,W)\cdot S(X,W)$ with $ R(X,W)=W^{-1}\,\psi(W^{-1}(X-X_c(W)),\xi(Z)),\qquad S(X,W)=\mathrm e^{\mathrm i\phi(X,Z)} $ one gets $ \partial_{11}^2U=\partial_{11}^2R\cdot S+2\partial_{1}R\cdot\partial_1S+R\cdot\partial_{11}^2S, $ with $ \partial_{1}R=W^{-2}\,\partial_{1}\psi(W^{-1}(X-X_c(W)),\xi(Z)), $ $ \partial_{11}^2R=W^{-3}\,\partial_{11}^2\psi(W^{-1}(X-X_c(W)),\xi(Z)), $ and $ \partial_1S=\mathrm i\partial_1\phi\cdot S,\qquad\partial_{11}^2S=(\mathrm i\partial_{11}^2-(\partial_1\phi)^2)\cdot S. $