### D.53 In­te­gral con­straints

This note ver­i­fies the men­tioned con­straints on the Coulomb and ex­change in­te­grals.

To ver­ify that , just check their de­f­i­n­i­tions.

The fact that

is real and pos­i­tive is self-ev­i­dent, since it is an in­te­gral of a real and pos­i­tive func­tion.

The fact that

is real can be seen by tak­ing com­plex con­ju­gate, and then not­ing that the names of the in­te­gra­tion vari­ables do not make a dif­fer­ence, so you can swap them.

The same name swap shows that and are sym­met­ric ma­tri­ces; and .

That is pos­i­tive is a bit trick­ier; write it as

with . The part within paren­the­ses is just the po­ten­tial of a dis­tri­b­u­tion of charges with den­sity . Sure, may be com­plex but that merely means that the po­ten­tial is too. The elec­tric field is mi­nus the gra­di­ent of the po­ten­tial, , and ac­cord­ing to Maxwell’s equa­tion, the di­ver­gence of the elec­tric field is the charge den­sity di­vided by : . So and the in­te­gral is

and in­te­gra­tion by parts shows it is pos­i­tive. Or zero, if is zero wher­ever is not, and vice versa.

To show that , note that

is non­neg­a­tive, for the same rea­sons as but with re­plac­ing . If you mul­ti­ply out the in­ner prod­uct, you get that is non­neg­a­tive, so .