A sigh is defined as a kind of paralinguistic respiration in the form of a deep, audible, single exhalation of air out of the mouth or nose.

Sighing may have purely physiological triggers (see: Respiratory variability preceding and following sighs: A resetter hypothesis ^{[ paywalled ]} in: Biological Psychology, Volume 84, Issue 1, Pages 82-87), or may arise from negative emotion, i.e. from feelings such as dismay, dissatisfaction, boredom, or futility.

According to a 2014 study in the Psychophysiology journal :

"How dimensions of emotion affect respiratory regulation assessed by respiratory variability and sighing is unknown."

There are also theories that sighs could be functioning as signals to convey emotional states to others :

[they are] often unintentional expressions of an activity, plan or desire that has to be discarded, creating a pause before it can be replaced by a novel initiative."

Source : Scandinavian Journal of Psychology ^{[ paywalled ]}

and :

[…] in contrast to largely involuntary respiratory acts like yawning or sneezing, sighing is wholly manipulable, which suggests that its occurrence in interaction may be purposeful."

Source : Berkeley Linguistics Society Journal, 29

Further extensive technical information The integrative role of the sigh in psychology, physiology, pathology, and neurobiology Progress in brain research. 2014;209:91-129.

Also see : Yawningplugin-autotooltip__plain plugin-autotooltip_bigYawning

The physical and psychological sequences which occur during the yawning process have been widely studied. Nevertheless, the reasons for yawning are disputed and unclear - as is identifying an evolutionary reason to explain the need for it.

" and Respiratory rhythmplugin-autotooltip__plain plugin-autotooltip_bigRespiratory rhythm

The neural regulators and feedback mechanisms which control the rhythmic nature of breathing in mammals (and many other organisms) are currently the subject of intense research.

"Mammalian central pattern generators producing rhythmic movements exhibit robust but flexible behavior.