Complex Number's Assignment (From my signals class)

[Naeem]

Q1. If j = square root of - 1 what is square root of j ?


what I did was : Plug in the value of j in square root of j and came up with -1 to the power of 0.25.

Is this right. Looks like this wrong.

Q2. Given a complex number w = x + jy, the complex congugate of w is defined in rectangular coordinates as w* = x-jy. Use this fact to derive complex congugation in polar form.


What I did was : multiply both w * w* and came up with x ^ 2 + y ^2

and I know euler's formula e ^(jtheta) = cos (theta) + i sin (theta)

Is this right, probably not, can someone guide me here as well.

Q3. By hand sketch the following against independent variable t:

(a) x2(t) = I am (3 - e(1-j2pi)t)


There is another two also in these parts, One with the real part given and another one with x3(t) = 3 - Im(e(1-j2pi)t)

How do I do these problems ? Please anyone help me. The book is worthless. It just talks about basics on complex numbers, congugates , polar forms etc.

But this HW has been a pain believe me...

 

[rbj]

Q1. If j = square root of - 1 what is square root of j ?what I did was : Plug in the value of j in square root of j and came up with -1 to the power of 0.25.

Is this right. Looks like this wrong.

i think your prof wants to know what the real part and imaginary part of $\sqrt{j}$ is. (maybe magnitude and angle, but that would be too easy.) remember, the square root of anything has two solutions.

Q2. Given a complex number w = x + jy, the complex congugate of w is defined in rectangular coordinates as w* = x-jy. Use this fact to derive complex congugation in polar form.What I did was : multiply both w * w* and came up with x ^ 2 + y ^2

and I know euler's formula e ^(jtheta) = cos (theta) + i sin (theta)

Is this right, probably not, can someone guide me here as well.

you are being asked to express $w$ in polar form and $w^*$ in terms of $w$ in polar form. i think that is the case.

Q3. By hand sketch the following against independent variable t:

(a) x2(t) = I am (3 - e(1-j2pi)t)There is another two also in these parts, One with the real part given and another one with x3(t) = 3 - Im(e(1-j2pi)t)

How do I do these problems ? Please anyone help me. The book is worthless. It just talks about basics on complex numbers, congugates , polar forms etc.

But this HW has been a pain believe me...

what do you know about the exponential? what happens when the exponent is the sum of two terms? what do you do with it?

try using $\LaTeX$ to express your math here on this forum. we ain't USENET here where you need to rely on "ASCII math".

 

[ravenprp]

\[ \sum_{k=1}^n k^2 = \frac{1}{2} n (n+1).\]

 

[Corneo]

For Q2: Say your given $w = x + jy$. Arbitrarily draw this on the x-y plane. Now draw $w^*$. Express w in polar form. What do you notice about w* in polar form?

 

[Corneo]

For Q1: One way to solve this is express j in polar form. You can get this from Euler's identiy and see that $e^{j \pi /2} = \cos (\pi/2)+ j \sin (\pi/2) = j$. Then you have

$$\sqrt{j} = \sqrt{e^{j \pi /2}}$$