# Image of a line

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Hi guys!

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!

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!

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#2

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.

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(Original post by

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.

*******deadness**)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.

Ah so are you saying I should substitute z=x into this?

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#4

Yeah, and I expect after simplifying the e bit and separating the whole thing into the real and imaginary parts you'll have something to work with

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(Original post by

Yeah, and I expect after simplifying the e bit and separating the whole thing into the real and imaginary parts you'll have something to work with

*******deadness**)Yeah, and I expect after simplifying the e bit and separating the whole thing into the real and imaginary parts you'll have something to work with

Getting ( root2/2+root2/2i)(x-i)+i

I have no clue what to do next after simplifying that

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#6

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.

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(Original post by

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.

*******deadness**)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.

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*******deadness**)

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.

However I am not sure what to do next

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#9

(Original post by

Hi guys!

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!

**maths4life2020**)Hi guys!

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!

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

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(Original post by

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

**mqb2766**)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

Sorry this question I've been working on all day so I have been going round in circles

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#11

(Original post by

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

**maths4life2020**)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

offset + half line at 45 degrees

So just work out the offset (rotation of the orign).

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(Original post by

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).

**mqb2766**)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).

As working it out to get i=ie^pi/4i

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#13

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(Original post by

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.

**mqb2766**)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.

Ah I'll keep on working on it!

Thanks for the help

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#15

(Original post by

The angle of rotation?

Ah I'll keep on working on it!

Thanks for the help

**maths4life2020**)The angle of rotation?

Ah I'll keep on working on it!

Thanks for the help

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(Original post by

Not sure what you don't understand?

**mqb2766**)Not sure what you don't understand?

My answers have been all over the place hahaha

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#17

(Original post by

I just don't get how to get the image of the line when arg(z)=0

My answers have been all over the place hahaha

**maths4life2020**)I just don't get how to get the image of the line when arg(z)=0

My answers have been all over the place hahaha

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

Which part don't you understand? I've left the offset (rotation of origin) for you to calculate.

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(Original post by

Copied from #9,

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

Which part don't you understand? I've left the offset (rotation of origin) for you to calculate.

**mqb2766**)Copied from #9,

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

Which part don't you understand? I've left the offset (rotation of origin) for you to calculate.

I get ie^pi/4i=i

Do I then solve this?

To get -1+root2

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#19

(Original post by

From calculating the offset

I get ie^pi/4i=i

Do I then solve this?

To get -1+root2

**maths4life2020**)From calculating the offset

I get ie^pi/4i=i

Do I then solve this?

To get -1+root2

How?

ie^pi/4i = e^i3pi/4

As i = e^ipi/2

If you're unsure, just do some simple geometry.

Last edited by mqb2766; 1 month ago

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I completely forgot about i = e^ipi/2

My brain wasn't working hahahaha

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