In [1, p. 60 exercise 3.3] we are asked to prove that the complex multivariate Gaussian PDF reduces to the complex univariate Gaussian PDF if N=1.
Solution: The complex multivariate Gaussian PDF is given by [1, p. 44, (3.13)] :
 $p(\mathbf{x_{N}})=\frac{1}{\pi^{N}det(\mathbf{C}_{xx})}exp\left[-(\mathbf{x_{N}}-\mathbf{\mu}_{\mathbf{x}_{N}})^{H}\mathbf{C}^{-1}_{xx}(\mathbf{x_{N}}-\mathbf{\mu}_{\mathbf{x}_{N}})\right]$ (1)

with $\mathbf{x_{N}} = \mathbf{u_{N}}+j\mathbf{v_{N}}$ denoting an $N$ dimensional complex vector, which is composed of the real $N$ dimensional vectors $\mathbf{u}_{N}$ and $\mathbf{v}_{N}$ . The univariate complex gaussian probability density function is given by [1, p. 44, (3.10)] :
 $p(x_{1})=\frac{1}{\pi \sigma_{x_{1}}^{2}} exp\left(- \frac{\left|x_{1} -\mathbf{\mu}_{x_{1} }\right|^{2}}{\sigma_{x_{1}}^{2}}\right)$ (2)

with $\sigma_{x_{1}}^{2}= E\left\{(x-\mu_{x})(x-\mu_{x})^{*}\right\}$ where $\mathbf{x}_{1}$ is a one dimensional complex variable with $x_{1}=u_{1} +j v_{1}$, $v_{1}$ and $u_{1}$ being real variables. For $N=1$ in (1) we obtain for the covariance matrix $\mathbf{C}_{xx}$ the following relation:
 $\mathbf{C}_{xx}$ $=$ $E\left\{(\mathbf{x_{N}}-\mathbf{\mu}_{\mathbf{x}_{N}})(\mathbf{x_{N}}-\mathbf{\mu}_{\mathbf{x}_{N}})^{H} \right\}$ $=$ $E\left\{(x_{1}-\mu_{x_{1}})(x_{1}-\mu_{x_{1}})^{H} \right\}$ $=$ $E\left\{(x_{1}-\mu_{x_{1}})(x_{1}-\mu_{x_{1}})^{*} \right\}$ $=$ $\sigma_{x}^{2}$

and thus $\mathbf{C}_{xx}^{-1}= \frac{1}{ \sigma_{x}^{2}}$. The determinant of an single element matrix is per definition the element value itself ([2, p. 326]), thus $\det(\mathbf{C}_{xx})=\sigma^{2}_{x}$ for $N=1$. Replacing $\mathbf{C}_{xx},\mathbf{C}_{xx}^{-1}$ and $\det(\mathbf{C}_{xx})$ in (1) while setting $N=1$ we obtain:
 $p(\mathbf{x_{N=1}})$ $=$ $\frac{1}{\pi \sigma_{x}^{2}}exp\left[ -\frac{(x_{1}-\mu_{x_{1}})^{H}( x_{1}-\mu_{x_{1}})}{\sigma_{x}^{2}}\right]$ $=$ $\frac{1}{\pi \sigma_{x}^{2}}exp\left[-\frac{\left|x_{1} -\mathbf{\mu}_{x_{1} }\right|^{2}}{\sigma_{x_{1}}^{2}}\right]$

which is equal to (2). QED.

[1] Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”, Prentice Hall, ISBN: 0-13-598582-X.
[2] Lawrence J. Corwin and Robert H. Szczarba: “Calculus in Vector Spaces”, Marcel Dekker, Inc, 2nd edition, ISBN: 0824792793.