Complex conjugate of a complicated function

In summary, the complex conjugate of a complex number z is z* = x - iy, and the complex conjugate of a complex function f(z) is f(z)* = u(x,y) - iv(x,y). For complex numbers on a circle, z* = exp(-iΘ) - i. While it may seem like the process of conjugation is simply replacing i with -i, this is not always the case. For more complicated functions, the process may involve converting them to the form a + ib first. However, there are some cases, such as (1/z)* = 1/(z*), where this conversion is not necessary.
  • #1
BomboshMan
19
0
Hi,

I know if we have a complex number z written as z = x +iy , with a and real, the complex conjugate is z* = x - iy. Also if we write a complex function f(z) = u(x,y) + iv(x,y), with u and v real valued, then similarly the complex conjugate of this function is f(z)* = u(x,y) - iv(x,y).

And if we have some complex numbers around a circle like z = i + exp(iΘ) then the conjugate of these numbers is z* = exp(-iΘ) - i.

It always seems like the way to conjugate something complex is to literally just 'put a minus in front of the i's ' ...but is this always the case?

Surely if we had some complicated function like g(x + iy) = [sin(x+iy) + iexp(cos(x-y))]^(-1) , we couldn't just replace the i's with (-i)'s? Seems like we'd have to convert the above to the form a + ib first.

If not - if it really is that elegant - is there a proof or something that it works in every case?

Thanks,

Matt
 
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  • #2
BomboshMan said:
Surely if we had some complicated function like g(x + iy) = [sin(x+iy) + iexp(cos(x-y))]^(-1) , we couldn't just replace the i's with (-i)'s?
Right.

Seems like we'd have to convert the above to the form a + ib first.
Not always.

As an example, (1/z)* = 1/(z*), you don't have to calculate real and imaginary parts of 1/z here.
 

FAQ: Complex conjugate of a complicated function

1. What is the definition of a complex conjugate?

A complex conjugate of a complex number is the number with the same real part but an opposite imaginary part. For example, the complex conjugate of 3+4i is 3-4i. It can also be thought of as reflecting the number across the real axis on the complex plane.

2. How do you find the complex conjugate of a complex function?

To find the complex conjugate of a complex function, replace all instances of i with -i. This will result in a new function that is the complex conjugate of the original.

3. What is the importance of the complex conjugate in mathematics?

The complex conjugate is important in mathematics because it allows for the simplification of complex expressions and equations. It also plays a role in finding roots and solutions to complex equations.

4. Can a complex function have more than one complex conjugate?

No, a complex function can only have one complex conjugate. This is because the complex conjugate is unique and is defined as the number with the same real part but an opposite imaginary part.

5. How does the complex conjugate affect the properties of a complex function?

The complex conjugate does not affect the properties of a complex function, as it is simply a reflection of the original function on the complex plane. However, it can be useful in finding symmetries and patterns in the behavior of a complex function.

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