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%% Branches of Complex Logarithm % %% % Create two Cartesian grids for X and Y in the interval |[-4,4]|. % Copyright 2015 The MathWorks, Inc. [X,Y] = meshgrid(-4:0.25:4,-4:0.25:4); %% % Calculate the complex logarithm on the grid. Use |1i| for improved speed % and robustness with complex arithmetic. Z = log(X + 1i*Y); %% % Make a surface plot of the imaginary portion of the function. surf(X,Y,imag(Z)) grid on hold on xlabel('X') ylabel('Y') zlabel('Z') title('Principal Branch of Im[$\log(X+iY)$]','Interpreter','latex') view(40,40) %% % On the complex plane, the natural logarithm is a multivalued function % that winds around the origin. %% % To obtain a different branch of the function, add $2\pi i$ to % the |Z| values. z2 = Z + 2*pi*1i; surf(X,Y,imag(z2)) title('Two Branches of the Complex Logarithm') %% % In this plot, the branches are stacked on top of each other and meet along % the negative real axis.