Complex numbers.

**Showcase_22** · October 2nd 2008, 10:06 AM

The question is:

Let $z=\frac{\pi}{6}+iln2$ . Write $e^{iz}$ in the form a+bi.

(careful! This is a trick question.)

What I did:

$e^{iz}=e^{\frac{i \pi}{6}-ln2}$

$e^{iz}=e^{\frac{i\pi}{6}}e^{-ln2}$

$e^{iz}=\frac {1}{2} e^{\frac{i\pi}{6}}$

$e^{iz}=(\frac{1}{2})(cos(\frac{\pi}{6}) + i sin(\frac{\pi}{6}))$

$e^{iz}=(\frac{1}{2})(\frac{\sqrt 3}{2}+ i \frac{1}{2})$

$e^{iz}=(\frac{1}{4})(\sqrt3+ i)$

I think that's right. It's just that when a lecturer has taken time out to write "this is a trick question" next to it, it implies it is much harder than it looks.

So is this the right answer?

**Prove It** · October 2nd 2008, 06:54 PM

Originally Posted by Showcase_22

The question is:

Let $z=\frac{\pi}{6}+iln2$ . Write $e^{iz}$ in the form a+bi.

(careful! This is a trick question.)

What I did:

$e^{iz}=e^{\frac{i \pi}{6}-ln2}$

$e^{iz}=e^{\frac{i\pi}{6}}e^{-ln2}$

$e^{iz}=\frac {1}{2} e^{\frac{i\pi}{6}}$

$e^{iz}=(\frac{1}{2})(cos(\frac{\pi}{6}) + i sin(\frac{\pi}{6}))$

$e^{iz}=(\frac{1}{2})(\frac{\sqrt 3}{2}+ i \frac{1}{2})$

$e^{iz}=(\frac{1}{4})(\sqrt3+ i)$

I think that's right. It's just that when a lecturer has taken time out to write "this is a trick question" next to it, it implies it is much harder than it looks.

So is this the right answer?

Well let's see...

If $z = x + iy$ then $e^z = e^x \ {\rm{cis}} (y)$ .

**Showcase_22** · October 2nd 2008, 10:42 PM

ahhhh! The last bit you wrote ended up with a Latex error!

**CaptainBlack** · October 2nd 2008, 10:46 PM

Originally Posted by Showcase_22

ahhhh! The last bit you wrote ended up with a Latex error!

Corrected, but what it says you knew already!

RonL

**CaptainBlack** · October 2nd 2008, 10:51 PM

Originally Posted by Showcase_22

The question is:

Let $z=\frac{\pi}{6}+iln2$ . Write $e^{iz}$ in the form a+bi.

(careful! This is a trick question.)

What I did:

$e^{iz}=e^{\frac{i \pi}{6}-ln2}$

$e^{iz}=e^{\frac{i\pi}{6}}e^{-ln2}$

$e^{iz}=\frac {1}{2} e^{\frac{i\pi}{6}}$

$e^{iz}=(\frac{1}{2})(cos(\frac{\pi}{6}) + i sin(\frac{\pi}{6}))$

$e^{iz}=(\frac{1}{2})(\frac{\sqrt 3}{2}+ i \frac{1}{2})$

$e^{iz}=(\frac{1}{4})(\sqrt3+ i)$

I think that's right. It's just that when a lecturer has taken time out to write "this is a trick question" next to it, it implies it is much harder than it looks.

So is this the right answer?

Well it looks OK, but we can just do a numerical check:

Code:

>I
                       0+1i 
>z=pi/6+I*log(2)
         0.523599+0.693147i 
>exp(I*z)
             0.433013+0.25i 
>
>
>(0.25)*(sqrt(3)+I)
             0.433013+0.25i 
>

RonL

**Showcase_22** · October 2nd 2008, 11:12 PM

Corrected, but what it says you knew already!

Actually, i'm not sure I do :s

What does $e^z=e^x cis (y)$ mean? More specifically, what does "cis" stand for?

Well it looks OK, but we can just do a numerical check:

>exp(I*z)
0.433013+0.25i

How did you work out that bit? Can your calculator work out complex numbers?

**Prove It** · October 3rd 2008, 07:27 PM

Originally Posted by Showcase_22

Actually, i'm not sure I do :s

What does $e^z=e^x cis (y)$ mean? More specifically, what does "cis" stand for?

$cis(y)$ is a shorthand way of writing $\cos{y} + i\sin{y}$ , which is a complex number of magnitude 1 written in polar co-ordinates.

By definition of $z = x + iy$ , we have

$e^z = e^{x+iy} = e^x e^{iy} = e^x cis(y)$

because, by Euler's formula

$e^{i\theta} = cis(\theta)$

**CaptainBlack** · October 3rd 2008, 10:19 PM

Originally Posted by Showcase_22

Actually, i'm not sure I do :s

What does $e^z=e^x cis (y)$ mean? More specifically, what does "cis" stand for?

It actualy means very little, it is a relativly recently invented notation for:

$\cos(x)+{\bold{i}}\sin(x)$

but such a notation is unneccessary, and redundant.

How did you work out that bit? Can your calculator work out complex numbers?

Well mine can, yes

RonL

**Showcase_22** · October 4th 2008, 01:55 AM

oh right!

That cis notation looks redundant but it is shorter than writing $cos x+i sinx$ . I might start using it on my whiteboard since I always run out of space on it!

My calculator can't do complex numbers. Well I don't think it can!

Does a TI-84 have a complex number setting? I haven't really used it and I didn't take the manual with me to university.

Also, for a trick question, that wasn't that bad. I guess lecturers write that there to freak students out!

Thread: Complex numbers.

LinkBack

Thread Tools

Display

Complex numbers.

Similar Math Help Forum Discussions

raising complex numbers to complex exponents?

Finding real numbers in equations containing complex numbers

What is complex slope of a line in complex numbers

Pre-Calc Graphing with Complex numbers. Multiplying with complex numbers.

Complex Numbers- Imaginary numbers

Search Tags