I have a question, if a complex function corresponds to a positive rotation about the point i by pi/4 radians. I get that the overall transformation is (z-i)*(e^pi/4*i)+i
But I don't understand how to find the image of the line arg(z)=0 under f(z)?
What method can I use to do it algebraically? I'm getting the answer as Re(z)
I think arg(z)=0 means z is a positive real number, and Im(z)=0, so instead of the usual z=x+yi, z=x in this case. Then you should be able to simplify it and match it to u+iv.
I think arg(z)=0 means z is a positive real number, and Im(z)=0, so instead of the usual z=x+yi, z=x in this case. Then you should be able to simplify it and match it to u+iv.
Thank you! Ah so are you saying I should substitute z=x into this?
I'd expand and match the real parts to u (real part of w) and match the imaginary parts to v (imaginary part of w) and see if I can form some relationship.
I'd expand and match the real parts to u (real part of w) and match the imaginary parts to v (imaginary part of w) and see if I can form some relationship.
I'd expand and match the real parts to u (real part of w) and match the imaginary parts to v (imaginary part of w) and see if I can form some relationship.
I have tried this and get u=root2/2(x+y) and w= root2/2(x-y)+1 However I am not sure what to do next
I have a question, if a complex function corresponds to a positive rotation about the point i by pi/4 radians. I get that the overall transformation is (z-i)*(e^pi/4*i)+i
But I don't understand how to find the image of the line arg(z)=0 under f(z)?
What method can I use to do it algebraically? I'm getting the answer as Re(z)
But honestly not sure with what I'm doing!
If anyone could help that you be great!
Can you not just use the transformation directly z = x*e^i0 So (z-i)*(e^pi/4*i)+i = z*(e^pi/4*i) + i-ie^pi/4*i = x*e^pi/4*i + offset
Can you not just use the transformation directly z = x*e^i0 So (z-i)*(e^pi/4*i)+i = z*(e^pi/4*i) + i-ie^pi/4*i = x*e^pi/4*i + offset
Oh thank you, I see. But how would I get the image of the line from this? Sorry this question I've been working on all day so I have been going round in circles
Oh thank you, I see. But how would I get the image of the line from this? Sorry this question I've been working on all day so I have been going round in circles
x is the non-negative modulus (real component of original line). The transformed line is offset + half line at 45 degrees So just work out the offset (rotation of the orign).
x is the non-negative modulus (real component of original line). The transformed line is offset + half line at 45 degrees So just work out the offset (rotation of the orign).
Would it be pi/4? As working it out to get i=ie^pi/4i
Would it be pi/4? As working it out to get i=ie^pi/4i
Would what be pi/4? You can do simple geometry to get the rotation of the origin by 45, or use the expression a couple of posts ago. It should be straightforward either way.
Would what be pi/4? You can do simple geometry to get the rotation of the origin by 45, or use the expression a couple of posts ago. It should be straightforward either way.
The angle of rotation? Ah I'll keep on working on it! Thanks for the help