What is the complete expansion of |1+a|^2 for a complex number a?

[cabrera]

Hi,

Could you help me to understand the following expansion I found in a book of qunatum mechanics.

|1+a|^2=1+a+a*+... where a* is the complex conjugate of a

 

[hilbert2]

What are the remaining terms in the expansion? I don't think it is an infinite series...

If $a=x+iy$, where $x$ and $y$ are real, then we have

$\left| 1+a \right|^{2} = (1+a)(1+a)^{*}=(1+x+iy)(1+x-iy)=1+x+x^{2}+y^{2}=1+\frac{a}{2}+\frac{a^{*}}{2}+a^{*}a$

 

[micromass]

What are the remaining terms in the expansion? I don't think it is an infinite series...

If $a=x+iy$, where $x$ and $y$ are real, then we have

$\left| 1+a \right|^{2} = (1+a)(1+a)^{*}=(1+x+iy)(1+x-iy)=1+x+x^{2}+y^{2}=1+\frac{a}{2}+\frac{a^{*}}{2}+a^{*}a$

I really don't see how you got this. To me it's

$$|1 + a|^2 = (1+a)(1+a)^* = (1 + a)(1 + a^*) = 1 + a + a^* + aa^*$$

 

[hilbert2]

^ Sorry, it should have been $1+2x+x^{2}+y^{2}$.

 

[Alpharup]

What are the remaining terms in the expansion? I don't think it is an infinite series...

If $a=x+iy$, where $x$ and $y$ are real, then we have

$\left| 1+a \right|^{2} = (1+a)(1+a)^{*}=(1+x+iy)(1+x-iy)=1+x+x^{2}+y^{2}=1+\frac{a}{2}+\frac{a^{*}}{2}+a^{*}a$

This expansion has errors...
If $a=x+iy$, where $x$ and $y$ are real, then we have

$\left| 1+a \right|^{2} = (1+a)(1+a)^{*}=(1+x+iy)(1+x-iy)=(1+x)^2+y^2=x^2+2x+1+y^2=1+x+iy+x-iy+(x+iy)(x-iy)=1+a+a{*}+(a)(a){*}$
Hope this helps...