3046.sagews

H(x) = 1/2*(psi(1-x/2)-psi(1/2-x/2))/gamma(1-x)

log(H(x))

log(-1/2*(psi(-1/2*x + 1/2) - psi(-1/2*x + 1))/gamma(-x + 1))

diff(log(H(x)),x,2)

-1/2*(2*(psi(-1/2*x + 1/2) - psi(-1/2*x + 1))*psi(-x + 1)/gamma(-x + 1) - (psi(1, -1/2*x + 1/2) - psi(1, -1/2*x + 1))/gamma(-x + 1))*gamma(-x + 1)*psi(-x + 1)/(psi(-1/2*x + 1/2) - psi(-1/2*x + 1)) + 1/4*(psi(1, -1/2*x + 1/2) - psi(1, -1/2*x + 1))*(2*(psi(-1/2*x + 1/2) - psi(-1/2*x + 1))*psi(-x + 1)/gamma(-x + 1) - (psi(1, -1/2*x + 1/2) - psi(1, -1/2*x + 1))/gamma(-x + 1))*gamma(-x + 1)/(psi(-1/2*x + 1/2) - psi(-1/2*x + 1))^2 + 1/4*(4*(psi(-1/2*x + 1/2) - psi(-1/2*x + 1))*psi(-x + 1)^2/gamma(-x + 1) - 4*(psi(1, -1/2*x + 1/2) - psi(1, -1/2*x + 1))*psi(-x + 1)/gamma(-x + 1) - 4*(psi(-1/2*x + 1/2) - psi(-1/2*x + 1))*psi(1, -x + 1)/gamma(-x + 1) + (psi(2, -1/2*x + 1/2) - psi(2, -1/2*x + 1))/gamma(-x + 1))*gamma(-x + 1)/(psi(-1/2*x + 1/2) - psi(-1/2*x + 1))

plot(H(x),(x,1,2))

p1=plot(gamma(x),(x,.25,4));
p2=plot(H(x),(x,-3,4), rgbcolor='green');
p=p1+p2;
p.show()

H1(x) = diff(H,x,1)

H2(x) = diff(H,x,2)

p1=plot(H(x), (x,1,2));
p2=plot(H1(x),(x,1,2),rgbcolor='red');
p3=plot(H2(x),(x,1,2),rgbcolor='green');
p=p1+p2+p3;
p.show()

LH(x) = log(H(x))

dLH(x) = diff(LH,x,1)

p1=plot(LH(x), (x,-5,10));
p2=plot(dLH(x),(x,-5,10),rgbcolor='red');
p=p1+p2;
p.show()

p1=plot(log(x+1), (x,-1,100));
p2=plot(dLH(x), (x,-1,100), rgbcolor='green');
p=p1+p2;
p.show()

p1=plot(LH(x),(x,-3,5));
p2=plot(1/12*(x-3/2)^3, (x,-3,5),rgbcolor='red');
p=p1+p2;
p.show()

p1=plot(LH(x), (x,-10,10));
p2=plot(dLH(x),(x,-10,10),rgbcolor='red');
p=p1+p2;
p.show()

plot(dLH(x),(x,1,2),rgbcolor='red');

x=var('x')
find_root(dLH(x)==0,1.2,1.4)

1.2703764753501168

x=var('x')
find_root(dLH(x)==0,1.8,1.9)

1.8375003327884909

n(LH(1.2703764753501168))

0.0149231702132196

n(LH(1.5))

0.00727650815941884

n(LH(1.8375003327884909))

-0.00856680957335963

plot(LH(x),(x,1,2))

x=var('x')
find_root(H1(x),1/2,3/2)

Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_21.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("eD12YXIoJ3gnKQpmaW5kX3Jvb3QoSDEoeCksMS8yLDMvMik="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single') File "", line 1, in <module> File "/tmp/tmpv2Nj4C/___code___.py", line 4, in <module> exec compile(u'find_root(H1(x),_sage_const_1 /_sage_const_2 ,_sage_const_3 /_sage_const_2 )' + '\n', '', 'single') File "", line 1, in <module> NameError: name 'H1' is not defined

x=var('x')
find_root(H1(x),3/2,5/2)

1.8375003327884789

plot(H1(x), (x,1,2))

d2LH=diff(LH(x),x,2)

p1=plot(LH(x), (x,-2,10));
p2=plot(dLH(x),(x,-2,40),rgbcolor='red');
p3=plot(d2LH(x),(x,-2,40),rgbcolor='green');
p=p1+p2+p3;
p.show()

dLH(x)

1/2*(2*(psi(-1/2*x + 1/2) - psi(-1/2*x + 1))*psi(-x + 1)/gamma(-x + 1) - (psi(1, -1/2*x + 1/2) - psi(1, -1/2*x + 1))/gamma(-x + 1))*gamma(-x + 1)/(psi(-1/2*x + 1/2) - psi(-1/2*x + 1))

plot(diff(H(x),x,3),(x,-3,5))

p1=plot(H(x), (x,-1,1));
p2=plot(H1(x),(x,-1,1),rgbcolor='red');
p3=plot(H2(x),(x,-1,1),rgbcolor='green');
p=p1+p2+p3;
p.show()

F(x) = x*H(x) + 1/gamma(1-x)

diff(F(x),x,1)

-1/2*(psi(-1/2*x + 1/2) - psi(-1/2*x + 1))*x*psi(-x + 1)/gamma(-x + 1) + 1/4*(psi(1, -1/2*x + 1/2) - psi(1, -1/2*x + 1))*x/gamma(-x + 1) - 1/2*(psi(-1/2*x + 1/2) - psi(-1/2*x + 1))/gamma(-x + 1) + psi(-x + 1)/gamma(-x + 1)

n(H(6.5))

280.300208881221

n(gamma(6.5))

287.885277815044

diff(1/gamma(1-x),x,1)

psi(-x + 1)/gamma(-x + 1)

x=var('x')
find_root(d2LH(x),8.1,9.5)

9.1950893711246984

plot(d2LH(x),(x,5,10),rgbcolor='green')

[1.5809660380138235,3.715227751138126,5.7414655980209677,7.7526800693048497; 3.1414677623024376,1,2,3]

Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_50.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("WzEuNTgwOTY2MDM4MDEzODIzNSwzLjcxNTIyNzc1MTEzODEyNiw1Ljc0MTQ2NTU5ODAyMDk2NzcsNy43NTI2ODAwNjkzMDQ4NDk3OyAzLjE0MTQ2Nzc2MjMwMjQzNzYsMSwyLDNd"),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single') File "", line 1, in <module> File "/tmp/tmprq5skb/___code___.py", line 3 [_sage_const_1p5809660380138235 ,_sage_const_3p715227751138126 ,_sage_const_5p7414655980209677 ,_sage_const_7p7526800693048497 ; _sage_const_3p1414677623024376 ,_sage_const_1 ,_sage_const_2 ,_sage_const_3 ] ^ SyntaxError: invalid syntax