# What is the derivative of this function?

Would any one tell me what is the $\partial^2U/\partial X^2$ where $$U(X,Z)=\frac{1}{W(Z)}\psi\left(\frac{X-X_c(Z)}{W(Z)},\xi(Z)\right)e^{i\phi(X,Z)}$$

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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 $$\partial_1S=\mathrm i\partial_1\phi\cdot S,\qquad\partial_{11}^2S=(\mathrm i\partial_{11}^2-(\partial_1\phi)^2)\cdot S.$$