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\}

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.