Let $h$ be a random variable and $g(h)$ be a real-valued function of $h$.


We know that if h is a real-random variable then:
$E\_h[g(h)] = \int\_{-\infty}^{\infty} f(h) g(h) dh$ where f(h) is the PDF of h.


I want to learn if there is an integral to express the expectation over $h$ when $h$ is a complex-random variable ($g(h)$ is still a real-valued function). The integral cannot be the same as the real-valued function as the limits of integral ($-\infty$ to $\infty$) do not cover the complex plain.



Integral form of expectation with respect to complex random variables
There is an unfortunate mismatch between the notations and concepts in probability used by mathematicians and by practitioners of many engineering fields. In the latter fields, it is conventional to refer to an expectation "with respect to" a particular random variable; in the former one may refer to $\operatorname E(g(H))$ where $H$ is a random variable, and hence $g(H)$ is a random variable. The expectation is defined as
$$
\operatorname E(g(H)) = \int\_\Omega g(H(\omega)) P(d\omega)
$$
where $\Omega$ is the set of all outcomes $\omega$ of a random experiment, each of which outcomes yields a particular value of $g(H),$ and $P$ is a measure that assigns probabilities $P(E)$ to sets $E\subseteq\Omega$ of outcomes.


Now first consider a simple case where, rather than spreading $g(H)$ about the whole complex plane $\mathbb C,$ we have $\Pr(H=1\text{ or } H=2 \text{ or } H=3)=1.$ Now the expected value is
\begin{align}
\operatorname E(g(H)) & = g(1)\Pr(H=1)+g(2)\Pr(H=2) + g(3)\Pr(H=3) \\[8pt]
& = \sum\_{h\,=\,1}^3 g(h)\Pr(H=h).
\end{align}
**Important:** Notice the way in which I distinguish between capital $H,$ the random variable, and lower-case $h,$ the possible value of $H.$ Without that distinction, even the simple expression $\Pr(H=h)$ cannot be understood. Mathematicians may perceive this present remark as a clarification to set a confused undergraduate straight, but I see books written by engineering professors that neglect this distinction and write probability mass functions and probability density functions as $f\_h(h)$ instead of $f\_H(h),$ and in fact that question posted above uses the lower-case $h$ for both of these two different things, where it ought instead to say
$$
\operatorname E(g(H)) = \int\_{-\infty}^{+\infty} g(h) f(h)\, dh
$$
with capital $H$ for the random variable and lower-case $h$ for the bound variable running through the set of all possible values of the random variable (capital) $H.$ The reason I'm writing this answer instead of telling you to post in some other place such as math(dot)stackexchange(dot)com or stats(dot)stackexchange(dot)com, as seems to be suggested by some votes to close the question, is that it is so often professors in engineering fields rather than confused undergraduates who make this mistake, and also mathematicians so seldom seem aware of the need to set this record straight.


If the value of $H$ is in $\mathbb C$ rather than $\mathbb R,$ then it is perfectly correct to say
$$
\operatorname E(g(H)) = \int\_\Omega g(H(\omega)) P(d\omega) = \int\limits\_\text{plane} g(h) f\_H(h) \, dh
$$
where $dh$ is the "element of area in the plane" and $f\_H(h)$ is the probability density of $H$ with respect to area in the plane. And here again, in the first integral I have capital $H$ and in the second lower-case $h$ except in the subscript, for a reason. Inattention to this distinction can actually render one unable to follow standard undergraduate textbook exercises.


