B735 CREW VIOLATED PROHIBITED AREA ON DEP.

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SHORTLY AFTER L TURN AT 400 FT; WENT IMC AND ENCOUNTERED MODERATE RIME ICING. TURNED ON ENG AND WING ANTI-ICE. WAS VERY BUSY DOING FLAP RETRACTION; AFTER TKOF; ETC. WHEN I SWITCHED TO DEP THEY INSTRUCTED US TO FLY HDG 280 DEGS. THEY SUBSEQUENTLY TURNED US TO HDG 310 DEGS. BEFORE THEY HANDED US OFF TO CTR; THEY GAVE ME THE PHONE NUMBER FOR TRACON AND INSTRUCTED ME TO CALL; AS A POSSIBLE INCURSION OF P-56 HAD OCCURRED. AFTER LEVELING OFF AT CRUISE ALT; I CONTACTED DISPATCH AND HAD THEM PATCH ME THROUGH TO DCA TRACON SUPVR. HE DID NOT ASK ANY QUESTIONS EXCEPT FOR MY NAME; ADDRESS; PHONE; AND AIRMAN'S CERTIFICATE NUMBER. HE ALSO TOLD ME HE WAS FILING A RPT THAT WE HAD VIOLATED P-56 AIRSPACE. I HAVE NO KNOWLEDGE; PERSONALLY; THAT WE IN FACT VIOLATED P-56. WE WERE; HOWEVER; GETTING INCORRECT INFO FROM OUR FLT DIRECTORS AND MY FO WAS HAVING TO FLY 'THROUGH' THE FLT DIRECTOR TO MANEUVER THE ACFT IN COMPLIANCE WITH THE SID AND ATC. I FEEL THAT THIS CIRCUMSTANCE PROBABLY CAUSED HIM TO MAINTAIN AN INADEQUATE ANGLE OF BANK TO AVOID P-56. I ACTUALLY HAD TO TURN OFF THE FLT DIRECTOR AND DIRECT MY FO TO FLY RAW DATA UNTIL WE COULD DETERMINE THE CAUSE OF ERRONEOUS OUTPUT FROM THE FLT DIRECTOR ONCE WE HAD SWITCHED OVER TO CTR. I TURNED BOTH FLT DIRECTOR MASTER SWITCHES OFF; THEN BACK ON; AND ENGAGED VNAV AND LNAV WHICH AT THAT TIME GAVE A PROPER CLB AND COURSE PATH; AND CONTINUED TO FUNCTION NORMALLY FOR THE REMAINDER OF THE FLT. IF WE DID IN FACT VIOLATE P-56 I WOULD SAY IT WAS COMPLETELY INADVERTENT; AND CAUSED BY A COMBINATION OF WX/ICING CONDITIONS; PLT WORKLOAD; MALFUNCTIONING FLT DIRECTORS; AND A SID WITH VERY LITTLE ROOM FOR ERROR. ALSO INVOLVED WAS PROBABLY TOO MUCH RELIANCE ON THE FLT DIRECTOR AND INADEQUATE OR INAPPROPRIATE INTERP OF RAW DATA. WE DID REVIEW AND DISCUSS THE SID TWICE PRIOR TO TKOF. THIS STATEMENT IS TRUE AND COMPLETE TO THE BEST OF MY KNOWLEDGE.