Calculating the Principal Value of (1+i)^-i | Quick Solution

[MissP.25_5]

Hello.
Can someone please check if I got the answer right? I don't have the right answers for my exercise so I need someone to check it for me, please.

Please find the principle value of (1+i)^-i

My attempt:
√2^(- i) = (e^ln√2)^(- i) = e^-i•ln√2
　　　　　　　　　　= cos(- ln√2) + i•sin(- ln√2)
　　　　　　　　　　= cos(ln√2) - i•sin(ln√2)


　　　　　　(1 + i) = √2[(1/√2) + i(1/√2)]
　　　　　　　　　　= √2[cos(π/4) + isin(π/4)]
　　　　　　　　　　= √2•e^i(π/4)


(1 + i)^(- i) = [√2•e^i(π/4)]^(- i)
　　　　　　　=[e^(π/4)] [√2^(- i)]
　　　　　　　=[e^(π/4)][cos(ln√2) - i•sin(ln√2)]


Therefore (1 + i)^(- i) = [e^(π/4)][cos(ln√2) - i•sin(ln√2)]

 

[SteamKing]

Hello.
Can someone please check if I got the answer right? I don't have the right answers for my exercise so I need someone to check it for me, please.

Please find the principle value of (1+i)^-i

My attempt:
√2^(- i) = (e^ln√2)^(- i) = e^-i•ln√2
　　　　　　　　　　= cos(- ln√2) + i•sin(- ln√2)
　　　　　　　　　　= cos(ln√2) - i•sin(ln√2)


　　　　　　(1 + i) = √2[(1/√2) + i(1/√2)]
　　　　　　　　　　= √2[cos(π/4) + isin(π/4)]
　　　　　　　　　　= √2•e^i(π/4)


(1 + i)^(- i) = [√2•e^i(π/4)]^(- i)
　　　　　　　=[e^(π/4)] [√2^(- i)]
　　　　　　　=[e^(π/4)][cos(ln√2) - i•sin(ln√2)]


Therefore (1 + i)^(- i) = [e^(π/4)][cos(ln√2) - i•sin(ln√2)]

It's not clear what these calculations represent:

√2^(- i) = (e^ln√2)^(- i) = e^-i•ln√2
　　　　　　　　　　= cos(- ln√2) + i•sin(- ln√2)
　　　　　　　　　　= cos(ln√2) - i•sin(ln√2)

　　　　　　(1 + i) = √2[(1/√2) + i(1/√2)]
　　　　　　　　　　= √2[cos(π/4) + isin(π/4)]
　　　　　　　　　　= √2•e^i(π/4)

You seem to have trouble expressing (1+i) in polar form and in applying Euler's formula correctly.

Remember:

z = (x+iy) = r(cos θ + i sin θ) = r e^{i θ}

where

r = \sqrt{x^{2}+y^{2}}

and

θ = arctan (y/x)

 

[MissP.25_5]

It's not clear what these calculations represent:



You seem to have trouble expressing (1+i) in polar form and in applying Euler's formula correctly.

Remember:

z = (x+iy) = r(cos θ + i sin θ) = r e^{i θ}

where

r = \sqrt{x^{2}+y^{2}}

and

θ = arctan (y/x)

But what's wrong? r=√2 and θ=∏/4.