logarithmic
Aug23-08, 10:49 AM
Sorry I forgot the out a title, and now I can't edit one in. :(
1. The problem statement, all variables and given/known data
Find all the solutions to \cos{z}=4i
2. Relevant equations
3. The attempt at a solution
What I did was write cos in terms of expoential, i.e. (1/2)(e^iz + e^-iz)=4i, and solved this quadratic to get e^{iz}=(4\pm\sqrt{17})i. Then I took the complex log of both sides, splitting the +/- into 2 equations
z=\frac{1}{i}(\ln{|(4+\sqrt{17})i|}+i\operatorname{arg}{[(4+\sqrt{17})i}])=-i\ln{(4+\sqrt{17})}+\pi/2+2k\pi
and the other solution is
z=\frac{1}{i}(\ln{|(4-\sqrt{17})i|}+i\operatorname{arg}{[(4-\sqrt{17})i]})=-i\ln{(4-\sqrt{17})}-\pi/2+2k\pi
The problem I'm having this that both these equation seem to be missing some possible solutions for the original problem. For example, i\ln{(4+\sqrt{17})}+\frac{3\pi}{2} is a solution which is cannot be gotten from my solutions with any value for k.
Can anyone see why my "solution" isn't getting all the possible solutions, or help me fix it so that it can get all the solutions.
1. The problem statement, all variables and given/known data
Find all the solutions to \cos{z}=4i
2. Relevant equations
3. The attempt at a solution
What I did was write cos in terms of expoential, i.e. (1/2)(e^iz + e^-iz)=4i, and solved this quadratic to get e^{iz}=(4\pm\sqrt{17})i. Then I took the complex log of both sides, splitting the +/- into 2 equations
z=\frac{1}{i}(\ln{|(4+\sqrt{17})i|}+i\operatorname{arg}{[(4+\sqrt{17})i}])=-i\ln{(4+\sqrt{17})}+\pi/2+2k\pi
and the other solution is
z=\frac{1}{i}(\ln{|(4-\sqrt{17})i|}+i\operatorname{arg}{[(4-\sqrt{17})i]})=-i\ln{(4-\sqrt{17})}-\pi/2+2k\pi
The problem I'm having this that both these equation seem to be missing some possible solutions for the original problem. For example, i\ln{(4+\sqrt{17})}+\frac{3\pi}{2} is a solution which is cannot be gotten from my solutions with any value for k.
Can anyone see why my "solution" isn't getting all the possible solutions, or help me fix it so that it can get all the solutions.