Prove for all Z E C |e^{z}-1| [itex]\leq[/itex] e^{|z|} - 1 [itex]\leq[/itex] |z|e^{|z|} I think this has to be proven using the triangle inequality but not sure how. Please help. :) thanks
Once i expanded them i realized they looked exactly like the triangle inequality where the modulus of the summation of terms was less than or equal to the modulus of each term summed. i didnt try to conclude |e^z| = e^|z| from what i was reading i think they are equal when z is real and the inequality holds when z has some imaginary part. but not too sure...
Good. That's about it. Except that they aren't necessarily equal when z is real and negative either, yes? It's only clearly true if z is real and postive.