7. We define the logarithm of the (nonzero) complex number z by E(z) = ln(r) + i

Question

7. We define the logarithm of the (nonzero) complex number z by E(z) = ln(r) + i(arg(z) + 2rk), where In( ) is the natural logarithm (of a positive real number), arg(z) is the principal argu- ment of z (in the interval [0,2T)), and k is any integer. Observe that 2(z) is multiply valued. (a) Show that E(i) =+2k) i for any integer k. (See p. 184 in our book.) b) Compute £(1+i (c) Compute ((3). Note: When you see the natural logarithm of a real number in this exercise, for example In(3), it is the same In(3) that you learned about in precalculus and calculus. The natural logarithm is a real valued function with only one value. The new thing is e(3), while In(3) is the same, friendly function you know and love.

Explanation / Answer

(a)

z =i

=>|z|=1

arg(z) =/2

l(i) =ln(1) +i(/2  +2k)

=>l(i) =0 +i(/2  +2k)

=>l(i) =i(/2  +2k)

-------------------------------------------------

(b)

z =1+i

=>|z|=[12+12] =2

arg(z) =/4

l(1+i) =ln(2) +i(/4  +2k)

---------------------------------------------------

(c)

z =3

=>|z|=3

arg(z) =0

l(3) =ln(3) +i(0  +2k)

=>l(3) =ln(3) +i(2k)

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