▁However , ▁in ▁most ▁cases ▁the ▁functions ▁f ( x ) ▁can ▁be ▁written ▁in ▁terms ▁of ▁an ▁infinite ▁series ▁of ▁orthogonal ▁functions ▁such ▁as ▁, ▁and ▁then ▁the ▁definition ▁becomes ▁ ▁which ▁is ▁slightly ▁more ▁understand able . ▁The ▁integral ▁is ▁shown ▁to ▁be ▁a ▁functional ▁integral ▁with ▁a ▁capital ▁D . ▁Sometimes ▁it ▁is ▁written ▁in ▁square ▁brackets : ▁[ D f ] ▁or ▁D [ f ], ▁to ▁indicate ▁that ▁f ▁is ▁a ▁function . ▁ ▁Ex amples ▁Most ▁functional ▁integr als ▁are ▁actually ▁infinite , ▁but ▁then ▁the ▁limit ▁of ▁the ▁quot ient ▁of ▁two ▁related ▁functional ▁integr als ▁can ▁still ▁be ▁finite . ▁The ▁functional ▁integr als ▁that ▁can ▁be ▁evaluated ▁exactly ▁usually ▁start ▁with ▁the ▁following ▁Gaussian ▁integral : ▁ ▁By ▁function ally ▁differenti ating ▁this ▁with ▁respect ▁to ▁J ( x ) ▁and ▁then ▁setting ▁ ▁to ▁ 0 ▁this ▁becomes ▁an ▁exponential ▁multip lied ▁by ▁a ▁polynomial ▁in ▁f . ▁For ▁example , ▁setting ▁, ▁we ▁find : ▁ ▁where ▁a , ▁b ▁and ▁x ▁are ▁ 4 - dimensional ▁vectors . ▁This ▁comes ▁from ▁the ▁formula ▁for ▁the ▁propag ation ▁of ▁a ▁phot on ▁in ▁quantum ▁elect rod ynam ics . ▁Another ▁useful ▁integral ▁is ▁the ▁functional ▁delta ▁function : ▁ ▁which ▁is ▁useful ▁to ▁specify ▁constraints . ▁Function al ▁integr als ▁can ▁also ▁be ▁done ▁over ▁Gr ass mann - valu ed ▁functions ▁, ▁where ▁, ▁which ▁is ▁useful ▁in ▁quantum ▁elect rod ynam ics ▁for ▁calculations ▁involving ▁ferm ions . ▁ ▁Appro