Integrating Gaussians with complex arguments

[hideelo]

The integral I'm looking at is of the form

$$\int_\mathbb{C} dz \: \exp \left( -\frac{1}{2}K|z|^2 + \bar{J}z \right)$$

Where $K \in \mathbb{R}$ and $J \in \mathbb{C}$

The book I am following (Kardar's Statistical Physics of Fields, Chapter 3 Problem 1) asserts that by completing the square this becomes $Z \exp\left( \frac{- |J|^2}{2K} \right)$ where $Z = \int_\mathbb{C} dz \: \exp \left( -\frac{1}{2}K|z|^2 \right)$. I can't seem to reproduce this, and I think the trouble I'm running into arises from $|z|^2$ not being a square, but rather it involves conjugation as well. Therefore, I get the following

$$-\frac{1}{2}K|z|^2 + \bar{J}z = -\frac{1}{2}K\left( z\bar{z} -2 \frac{\bar{J}}{K}z \right)= -\frac{1}{2}K\left( z\bar{z} -2 \frac{\bar{J}}{K}z - 2 \frac{J}{K}\bar{z} +2 \frac{J}{K}\bar{z} + 4\frac{ |J|^2}{K^2} - 4 \frac{ |J|^2}{K^2} \right) =$$

$$-\frac{1}{2}K\left( z - 2 \frac{J}{K} \right) \left( \bar{z} -2 \frac{\bar{J}}{K} \right) -J\bar{z} +2 \frac{ |J|^2}{K}$$

Which means that I'm getting that

$$\int_\mathbb{C} dz \: \exp \left( -\frac{1}{2}K|z|^2 + \bar{J}z \right) = \left[ \int_\mathbb{C} dz \: \exp \left( -\frac{1}{2}K \left| z-2 \frac{J}{K} \right|^2 - J\bar{z} \right) \right] \exp\left( 2 \frac{ |J|^2}{K} \right)$$

Which doesn't at all seem like

$$\left[ \int_\mathbb{C} dz \: \exp \left( -\frac{1}{2}K|z|^2 \right) \right] \exp\left( \frac{- |J|^2}{2K} \right)$$

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[mathman]

I haven't worked through all the details, but it looks like there was a shift in $z$, i.e. $z'=z-\frac{2J}{K^2}$.

 

[hideelo]

I haven't worked through all the details, but it looks like there was a shift in $z$, i.e. $z'=z-\frac{2J}{K^2}$.

I don't think shifting $z$ by anything can help. Suppose you sent $z \mapsto z+a$ for any $a$ then I would get the following

$$-\frac{1}{2}K |z|^2+\bar{J}z \mapsto -\frac{1}{2}K |z+a|^2+\bar{J}(z + a) =$$
$$-\frac{1}{2}K \left( z\bar{z} + a\bar{z} + z\bar{a} +a\bar{a} -\frac{2\bar{J}}{K} z - \frac{2\bar{J}}{K}a \right) =$$
$$-\frac{1}{2}K \left( z\bar{z} + a\bar{z} + z \left( \bar{a} - \frac{2\bar{J}}{K} \right) +a\bar{a} - \frac{2\bar{J}}{K}a \right)$$

If I want this to look like $-\frac{1}{2}K|z+b|^2 + c$, then I need to add and subtract terms with $\bar{z}$ which means that I can't pull $e^c$ out of the integral.

 

[mathman]

I'm getting a similar problem. It looks it will work only if $J$ and $z$ are real.

 

[mathman]

If the original exponent had an additional term $+J\bar z$, it might work.