The sigma function of general curve of genus one

/*

For the genus 1 curve 

    y^2 + (mu1*x + mu3)*y = x^3 + mu2*x^2 + mu4*x + mu6,

first few terms of the expansion of the sigma function around the origin is given by 

*/

Sigma(u)={u
+(mu2+1/4*mu1^2)*u^3/3!
+(mu2^2+1/2*mu1^2*mu2+(1/16*mu1^4+mu3*mu1+2*mu4))*u^5/5!
+(mu2^3+3/4*mu1^2*mu2^2+(3/16*mu1^4+3*mu3*mu1+6*mu4)*mu2+(1/64*mu1^6+3/4*mu3*mu1^3+3/2*mu4*mu1^2+(6*mu3^2+24*mu6)))*u^7/7!
+(mu2^4+mu1^2*mu2^3+(3/8*mu1^4+6*mu3*mu1+12*mu4)*mu2^2+(1/16*mu1^6+3*mu3*mu1^3+6*mu4*mu1^2+(72*mu3^2+288*mu6))*mu2+(1/256*mu1^8+3/8*mu3*mu1^5+3/4*mu4*mu1^4+(9*mu3^2+72*mu6)*mu1^2-36*mu3*mu4*mu1-36*mu4^2))*u^9/9!
+(mu2^5+5/4*mu1^2*mu2^4+(5/8*mu1^4+10*mu3*mu1+20*mu4)*mu2^3+(5/32*mu1^6+15/2*mu3*mu1^3+15*mu4*mu1^2+(684*mu3^2+2736*mu6))*mu2^2+(5/256*mu1^8+15/8*mu3*mu1^5+15/4*mu4*mu1^4+(201*mu3^2+1368*mu6)*mu1^2-564*mu3*mu4*mu1-564*mu4^2)*mu2+(1/1024*mu1^10+5/32*mu3*mu1^7+5/16*mu4*mu1^6+(15/2*mu3^2+171*mu6)*mu1^4-141*mu3*mu4*mu1^3-141*mu4^2*mu1^2+(-36*mu3^3-144*mu6*mu3)*mu1+(-72*mu3^2-288*mu6)*mu4))*u^11/11!
+(mu2^6+3/2*mu1^2*mu2^5+(15/16*mu1^4+15*mu3*mu1+30*mu4)*mu2^4+(5/16*mu1^6+15*mu3*mu1^3+30*mu4*mu1^2+(6216*mu3^2+24864*mu6))*mu2^3+(15/256*mu1^8+45/8*mu3*mu1^5+45/4*mu4*mu1^4+(3183*mu3^2+18648*mu6)*mu1^2-5916*mu3*mu4*mu1-5916*mu4^2)*mu2^2+(3/512*mu1^10+15/16*mu3*mu1^7+15/8*mu4*mu1^6+(426*mu3^2+4662*mu6)*mu1^4-2958*mu3*mu4*mu1^3-2958*mu4^2*mu1^2+(-648*mu3^3-2592*mu6*mu3)*mu1+(-1296*mu3^2-5184*mu6)*mu4)*mu2+(1/4096*mu1^12+15/256*mu3*mu1^9+15/128*mu4*mu1^8+(75/16*mu3^2+777/2*mu6)*mu1^6-1479/4*mu3*mu4*mu1^5-1479/4*mu4^2*mu1^4+(-231*mu3^3-648*mu6*mu3)*mu1^3+(-738*mu3^2-1296*mu6)*mu4*mu1^2-828*mu3*mu4^2*mu1+(-552*mu4^3+(-216*mu3^4-1728*mu6*mu3^2-3456*mu6^2))))*u^13/13!
+O(u^15)}

\\ This is a Hurwitz integral series over Z[mu1/2, mu3, mu2, mu4, mu6]. 

The sigma function of general curve of genus two

/*

For the genus 2 curve 

    y^2 + (mu01*x^2 + mu03*x + mu05)*y = x^5 + mu02*x^2 + mu04*x^3 + mu06*x^2 + mu08*x + mu10,

first few terms of the expansion of the sigma function around the origin is given by 

*/

Sigma(u3,u1)={u3-2*u1^3/3!
+(-8*mu02-2*mu01^2)*u1^5/5!
+(-32*mu02^2-16*mu01^2*mu02-2*mu01^4-2*mu03*mu01-4*mu04)*u1^7/7!
+(-mu03*mu01-2*mu04)*u3*u1^4/4!
+(-128*mu02^3-96*mu01^2*mu02^2-24*mu01^4*mu02-48*mu03*mu01*mu02-96*mu04*mu02-2*mu01^6-12*mu03*mu01^3-24*mu04*mu01^2+32*mu05*mu01+16*mu03^2+64*mu06)*u1^9/9!
+((-4*mu03*mu01-8*mu04)*mu02-mu03*mu01^3-2*mu04*mu01^2-4*mu05*mu01-2*mu03^2-8*mu06)*u3*u1^6/6!
+(-2*mu05*mu01-1/2*mu03^2-2*mu06)*u3^2*u1^3/2!/3!
+(2*mu05*mu01+1/4*mu03^2+mu06)*u3^3/3!
+(-512*mu02^4-512*mu01^2*mu02^3-192*mu01^4*mu02^2-480*mu03*mu01*mu02^2-960*mu04*mu02^2-32*mu01^6*mu02-240*mu03*mu01^3*mu02-480*mu04*mu01^2*mu02+384*mu05*mu01*mu02+192*mu03^2*mu02+768*mu06*mu02-2*mu01^8*mu01^2-30*mu03*mu01^5*mu01^2-60*mu04*mu01^4*mu01^2+96*mu05*mu01^3*mu01^2-54*mu03^2*mu01^2+192*mu06*mu01^2-408*mu04*mu03*mu01+800*mu05*mu03-408*mu04^2+1600*mu08)*u1^11/11!
+(-16*mu03*mu01*mu02^2-32*mu04*mu02^2-8*mu02*mu01^3*mu03-16*mu02*mu01^2*mu04-32*mu02*mu01*mu05-16*mu02*mu03^2-64*mu02*mu06-mu01^2*mu01^5*mu03-2*mu01^2*mu01^4*mu04-8*mu01^2*mu01^3*mu05-5*mu01^2*mu03^2-16*mu01^2*mu06-4*mu04*mu03*mu01+-16*mu05*mu03+-4*mu04^2-32*mu08)*u3*u1^8/8!
+(-8*mu05*mu01-2*mu02*mu03^2-8*mu02*mu06-2*mu05*mu01^3-1/2*mu01^2*mu03^2-2*mu01^2*mu06-4*mu05*mu03-8*mu08)*u3^2*u1^5/2!/5!
+(-mu05*mu03-2*mu08)*u3^3*u1^2/3!/2!} \\ +...

\\ This is a Hurwitz integral series over Z[mu01, mu03/2, mu05, mu02, mu04, mu06, mu08, mu10].