# Problem 1

Define the function we will be using for the forward differencing as $u=ge^{ikx_j}$ $\frac{\partial{u}}{\partial{x}}=iku$ and the forward differencing equation as $\Delta_x^{'}f_j=\frac{f_{j+1}-f_j}{\Delta}$ where are function $$f$$ will be our functuon $$u$$.
Putting this is we get $\Delta_x^{'}u=\frac{ge^{ikx_{j+1}}-ge^{ikx_j}}{\Delta}$ define $$x_{j+1}=x_j+\Delta$$ then the above equation becomes $\Delta_x^{'}u=\frac{ge^{ikx_j+\Delta}-ge^{ikx_j}}{\Delta}$
Factoring we get $\Delta_x^{'}u=\frac{ge^{ikx_j}(e^{ik\Delta}-1)}{\Delta}$ and taylor expanding we get $\approx\frac{ge^{ikx_j}(1+ik\Delta+\frac{(ik\Delta)^2}{2}-1)}{\Delta}$ Then factoring $=\frac{ik\Delta ge^{ikx_j}(1+\frac{ik\Delta}{2})}{\Delta}$. then $\Delta_x^{'}u=\frac{\partial{u}}{\partial{x}}(1+\frac{ik\Delta}{2})$ So $\Delta_x^{'}=(1+\frac{ik\Delta}{2})\frac{\partial{}}{\partial{x}}$