Question

Compute the following values when the log is defined by its principal value on the open set U equal to the plane with the positive real axis deleted.

a. log i
b. log(-1)
c. log(-1 + i)
d. i^i
e. (-i)^i

Answer 1

Following are the answer to this question:

Explanation:

The principle vale of Arg(3)

`Arg(3)=-\pi+\tan^(-1) ((|Y|)/(|x|))`

The principle value of the
`\logi= \log(0+i)\ \ \ \ \ _(where) \ \ \ x=0 \ \ y=1> 0`

So, the principle value:

a)

`\to \log(i)=\log |i|+i Arg(i)\\\\`

`=\log √(0+1)+i \tan^(-1)((1)/(0))\\\=\log 1 +i \tan^(-1)(\infty)\\\=0+i(\pi)/(2)\\\=i(\pi)/(2)`

b)

`\to \log(-i)= \log(0-i ) \ \ \ x=0 \ \ \ y= -1<0\\`

Principle value:

`\to \log(-i)= \log|-i|+iArg(-i) \\\\`

`=\log √(0+1)+i(-\pi+\tan^(-1)(\infty))\\\\=\log1 + i(-\pi+(\pi)/(2))\\\\=-i(\pi)/(2)`

c)

`\to \log(-1+i) \ \ \ \ x=-1, _(and) y=1 \ \ \ x<0 and y>0`

The principle value:

`\to \log(-1+i)=\log |-1+i| + i Arg(-1+i)`

`=\log √(1+1)+i(\pi+\tan^(-1)((1)/(1)))\\\\=\log √(2) + i(\pi-\tan^(-1)(\pi)/(4))\\\\=\log √(2) + i\tan^(-1)(3\pi)/(4)\\\\`

d)

`\to i^i=w\\\\w=e^(i\log i)`

The principle value:

`\to \log i=i(\pi)/(2)\\\\\to w=e^{i(i (\pi)/(2))}\\\\=e^{-(\pi)/(2)}`

e)

`\to (-i)^i\\\to w=(-i)^i\\\\w=e^(i \log (-i))`

In this we calculate the principle value from b:

so, the final value is
`e^{(\pi)/(2)}`

f)

`\to -1^i\\\\\to w=e^(i log(-1))\\\\\ principle \ value: \\\\\to \log(-1)= \log |-1|+iArg(-i)`

`=\log √(1) + i(\pi-\tan^(-1)(0)/(-1))\\\\=\log √(1) + i(\pi-0)\\\\=\log √(1) + i\pi\\\\=0+i\pi\\=i\pi`

and the principle value of w is =
`e^(\pi)`

g)

`\to -1^(-i)\\\\\to w=e^(-i \log (-1))\\\\`

from the point f the principle value is:

`\to \log(-1)= i\pi\\\to w= e^(-i(i\pi))\\\\\to w=e^(\pi)`

h)

`\to \log(-1-i)\\\\\ Here x=-1 ,<0 \ \ y=-1<0\\\\ \ principle \ value \ is:\\\\ \to \log(-1-i)=\log√(1+1)+i(-\pi+\tan^(-1)(1))`

`=\log√(2)+i(-\pi+(\pi)/(4))\\\\=\log√(2)+i(-(3\pi)/(4))\\\\=\log√(2)-i(3\pi)/(4))\\`