The derivation of the word pranayama- means the dimensions of the breath. Prana+ang+yam+ghan= pranayama. The word pranayama is derived from the suffix ghn from the verb yama- samyamane.

Yoga Darshan-

After the asana is accomplished in the eight limbs of yoga, pranayama is performed sequentially. Maharishi Patanjali explains the characteristics of pranayama:

tasmin sati shvasaprashvasyorgativiched: pranayam:|| 2.49||

That is, when the asana is perfected, the interruption of the movement of breathing = breaking the order = that is, breaking the flow of the going:- cutting is the pranayama. That is, pranayama is the distinction (interruption) of the natural movement of breathing.

Agni Purana-

nasikaaputamangulya'pidyaiv paren ch| 
audaram rechayedvayum rechanadrechak: smrut:||7|| 
bahyen vayuna deham drutivatpurayedyatha| 
tatha purnasch santishthetpuranatpurak: smrut:||8||

na munchati na gruöaati vayumantarbahi: sthitam| 
sampurnkumbhavattishthedachal: sa tu kumbhak:||9||

The air residing within the body is called prana. The name of stopping him is dimension. Therefore, pranayama means to stop the breath. His method is as follows: Press one hole of the nose with your finger and expel the air in the stomach through the other hole. This action is called laxative because it causes digestion. Then, like a bellows, fill the body with outside air and hold it for some time, this action is called supplementation. When the air within is not released, nor the air outside is taken in, but remains motionless like a perfect pot, it is called a pot.

Skanda Purana-

evam sansadhya niyaman sayamanshch vichakshan:| 
pranayamay sandadhyannanyatha yogasadhak:||26||

yato'shuchisharirasya vayukopo mahanbhavet|

vayukopatkusthata ch jadtvadinupashnute||27||

tasmadvichakshan: shuddham krutva deham yatetparam| 
pranayamasya vakshyami lakshanam shrunu pandav||28|| 

pranaapananirodhasch pranayam: prakirtit:| 
laghumadhyottariyakhya: sa ch dhiraistridhodit:||29||

laghurdvadashamatrastu matra nimish unmish:| dviguno madhyamaschoktastrigunaschottam: smrut:||30||
A wise person should practice asanas and pranayama by following the rules of Yama. Otherwise, there is no progress in yoga practice. When one starts practicing yoga with an impure body, there is an outbreak of air. This causes leprosy and inertia and other disorders. Therefore, a wise person should engage in pranayama only after purification of the body. O Arjuna! Listen to the characteristics of pranayama: Pranayama is to stop the air called prana and apana. The patient people also call it threefold with the distinction of small, medium and good. A moment is called a moment. The twelve-matra pranayama is called short, medium when it is doubled (24 matra) and excellent when it is tripled (36 matra).

Vishnu Purana-

pranakhyamanilam vashyamabhyasatkurute tu yat| 
pranayamasya vignyeyassbijo'bij eva ch||40||

Pranayama is the act of subduing the air called prana through practice. There are two divisions of it, Sabiz and Abiz.
parasparenaabhibhavam pranaapanau yathanilau| kurutassadvidhanen trutiyassanyamattayo:||41||
When the yogi restrains the prana and apana airs from each other while doing pranayama under the teachings of the Sadhguru, then the two pranayama named puraka and rechaka are restrained and when he combines the two together, the third kumbhaka pranayama occurs.

Mahashiva Purana-

mantrayog: sparshayogo bhavayogastathapar:| abhavayog: sarvebhyo mahayog: paro mat:||7||

Mantra yoga, sparsha yoga, abhava yoga and the greatest Maha yoga are considered.

mantrabhyasavashenaiv mantravachyarthagochar:| 
avyakshepaa manovruttirmantrayog udahrut:||8||

The instinct of the mantra which becomes visible in the verbal meaning of the mantra by the practice of the mantra is called mantra yoga.

pranayamamukha: saiv sparshayogo'bhidhiyate| 
sa mantrasparshanirmukto bhavayog: prakirtit:||9||

The same attitude with pranayama is called sparsha yoga. That same pranayama free from mantra-touch is called bhava yoga.

Brihannardiya Purana-

eshaamekatamam baddhava gurubhaktiparayan:|
upasako jayetpranan dvandvatito vimatsar:||116||
pramukhodamukho vapi tatha pratyamukho'pi va| 
abhyasen jayetprananni:shabde janavajareejate||117||
prano vayu: sharirasth aayamastasya nigrah:| 
pranayam iti prokto dvividh: sa prakeetrtit:||118||

The seeker should choose one of these asanas and practice it. This makes the seeker devoted to Gurubhakti life-conquering, transcendental and free from envy. In a secluded place, place asana facing east, north or west and practice continuously to control the breath.

The sages of the Chāndogya Upanishad say:

Prana is the father, Prana is the mother, Prana is the brother|
Prana is the sister, Prana is the teacher, Prana is the Brahmin|| (Chhandogi- 7.15.1)

Along with the physical changes, the emotional changes in the mind of a person through the various dimensions of yoga, i.e., prana, are as authentic and scientific as the physical changes.
For a person suffering from physical diseases like depression and schizophrenia, the sages of the yoga tradition say: O man! You take refuge in the soul, do pranayama. This is the life father, this life is the mother, this is the brother and sister. Prana is the Acharya and the Brahmin. Here the souls are described in metaphor.