- #1
benorin said:probably ought to start with [itex]\cos z = \frac{1}{2}\left(e^{iz}+e^{-iz}\right)[/itex], substituting into the given equation we have
[tex] \frac{1}{2}\left(e^{iz}+e^{-iz}\right)=2[/tex]
now multiply both sides by [itex]2e^{iz}[/itex] and collect all the terms on one side...
Nick O said:Very close! You're just missing one part to capture the periodic nature of cosine.
From ##e^{iz} = 2\pm \sqrt{3}\Rightarrow iz = \ln (2 \pm \sqrt{3})##, you can expand the RHS, taking into consideration the fact the complex logarithm is multiple valued.MissP.25_5 said:Which part? I don't understand. I thought that would be enough?
CAF123 said:From ##e^{iz} = 2\pm \sqrt{3}\Rightarrow iz = \ln (2 \pm \sqrt{3})##, you can expand the RHS, taking into consideration the fact the complex logarithm is multiple valued.
It is, the method is fine. Given ##z \in \mathbb{C},## what is ##\log_{e} z##?MissP.25_5 said:So how should I solve this question? I guess my working is alright up to this part, right?
lurflurf said:in fact by convention the principle value is
$$\mathrm{Cos}^{-1}(2)=\imath\, \mathrm{Cosh}^{-1}(2)=\imath\, \mathrm{Ln}(2+\sqrt{3})$$
CAF123 said:It is, the method is fine. Given ##z \in \mathbb{C},## what is ##\log_{e} z##?
It is not what I meant for you to write down - have you studied the complex logarithm?MissP.25_5 said:##\log_{e} z## = loge(x+iy)
Is that correct?
Same way you did.MissP.25_5 said:I don't get that part. How do you calculate that?
CAF123 said:It is not what I meant for you to write down - have you studied the complex logarithm?
lurflurf said:Same way you did.
$$\mathrm{Cos}^{-1}(2)=\imath\, \mathrm{Cosh}^{-1}(2)=\imath\, \mathrm{Ln}(2+\sqrt{3})\sim 1.316957896924816708625046347307968444026981971\imath
$$
is the principal value.
That is the agreed reference value we give when we give only one value.
To find all values you do not need to deal with complex logs (though you could).
Just think back to real trigonometry.
We found one value call it P
then -P was another
then we could add 2n ∏
Once we find one solution, we then find them all
what are all solutions to
$$\cos(x)=2$$
is done the same way as
what are all solutions to
$$\cos(x)=0.2$$
MissP.25_5 said:Yes, I have. But I don't know how to use that here.
lurflurf said:how would you do the one with 0.2?
It is the same.
If t is any solutions and k is an integer
$$2k\, \pi\pm t$$
is all the solutions because
$$\cos(2k\, \pi\pm t)=\cos(t)$$
by properties of cos(x)
CAF123 said:Ok, so you know that ##\log_e z = \text{Log}_e |z| + i\text{Arg}\,z + 2\pi k, \,\,k \in \mathbb{Z}##, where ##\log## is the complex log and ##\text{Log}## is the real log. Use this to obtain the full solution set of the equation ##iz = \ln(2\pm \sqrt{3})##.
Alternatively, as lurflurf said, make it clear that you solved for the principal value and, using the properties of the cosine, tag on a ##2\pi k.##
CAF123 said:Ok, so you know that ##\log_e z = \text{Log}_e |z| + i\text{Arg}\,z + 2\pi k, \,\,k \in \mathbb{Z}##, where ##\log## is the complex log and ##\text{Log}## is the real log. Use this to obtain the full solution set of the equation ##iz = \ln(2\pm \sqrt{3})##.
Alternatively, as lurflurf said, make it clear that you solved for the principal value and, using the properties of the cosine, tag on a ##2\pi k.##
lurflurf said:You are over thinking
if we know the solution
$$t=\imath\, \mathrm{Ln}(2+\sqrt{3})$$
to find all solutions we consider the additive inverse and shifts by 2 k pi
$$t=2k\, \pi+\imath\, \mathrm{Ln}(2\pm \sqrt{3})=2k\, \pi\pm\imath\, \mathrm{Ln}(2+ \sqrt{3})$$
lurflurf said:^Yes.
Do you intend you ln to be single valued, or multivalued?
cos has period 2pi so adding an integer multiple of 2pi does not change it
##Z_{1,2} = 2 \pm \sqrt{3}## are both real numbers > 0 so ##\text{Arg}\, Z_{1,2} = 0.##MissP.25_5 said:But how do I find Arg(Z) here? What is the value of r?
CAF123 said:##Z_{1,2} = 2 \pm \sqrt{3}## are both real numbers > 0 so ##\text{Arg}\, Z_{1,2} = 0.##
##\log_e z = w \iff z = e^w##, where ##\log_e## on the LHS is the complex logarithm. In your case, $$\log_e e^{iz} = \log_e (2\pm \sqrt{3}) \Rightarrow iz = \dots $$
Expand using the definition of the complex log.
That attachment is fine.MissP.25_5 said:Since the instruction says to find all solutions, doesn't that mean ln have to be multivalued? Multivalued means k>0, right? k=0 would be the principle value, which is single valued, isn't it?
Can you check the attachment?
lurflurf said:That attachment is fine.
It is a minor technicality. To find all solutions we can use a multivalued inverse, or we can use a single value inverse to generate all solutions. k=0 can be the principal value if it is set up that way
your last post is confusing
where did i^0 come from?
third line from the bottom should have 2k pi i
then it should in the next line become 2k pi when you multiply both sides by -i
lurflurf said:That attachment is fine.
It is a minor technicality. To find all solutions we can use a multivalued inverse, or we can use a single value inverse to generate all solutions. k=0 can be the principal value if it is set up that way
your last post is confusing
where did i^0 come from?
third line from the bottom should have 2k pi i
then it should in the next line become 2k pi when you multiply both sides by -i
lurflurf said:^Yes that looks good.