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Domdamo
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According to [Erdely A,1953; Higher Transcendental Functions, Vol I, Ch. VI.] the confluent hypergeometric equation
[tex]\frac{d^2}{d x^2} y + \left(c - x \right) \frac{d}{d x} y - a y = 0[/tex]
has got eight solutions, which are the followings:
[tex]y_1=M[a,c,x][/tex]
[tex]y_2=x^{1-c}M[a-c+1,2-c,x][/tex]
[tex]y_3=e^{x}M[c-a,c,-x][/tex]
[tex]y_4=x^{1-c}e^{x}M[1-a,2-c,-x][/tex]
[tex]y_5=U[a,c,x][/tex]
[tex]y_6=x^{1-c}U[a-c+1,2-c,x][/tex]
[tex]y_7=e^{x}U[c-a,c,-x][/tex]
[tex]y_8=x^{1-c}e^{x}U[1-a,2-c,-x][/tex]
where ##M[a,c,x]## is the confluent hypergeometric function of the first kind
and ##U[a,c,x]## is the confluent hypergeometric function of the second kind.
I would like to know how to evaluate exactly (step by step) the wronskian of the solutions. For example the ##W(y_1,y_5)## or ##W(y_5,y_7)##.
[tex] W(y_1,y_5) = y_{1} y_{5}^{'} - y_{1}^{'} y_{5} = ? =-\frac{\Gamma(c)}{\Gamma(a)} [/tex]
[tex] W(y_5,y_7) = y_{5} y_{7}^{'} - y_{5}^{'} y_{7} = ? = e^{j\cdot\pi\cdot \text{sign} \left[\Im(x)\right]\cdot (c-a)}[/tex]
The results are given in the referred book but the calculation is missing.
Can anybody help me or suggest a hint? Can anybody offer a article for this problem?
[tex]\frac{d^2}{d x^2} y + \left(c - x \right) \frac{d}{d x} y - a y = 0[/tex]
has got eight solutions, which are the followings:
[tex]y_1=M[a,c,x][/tex]
[tex]y_2=x^{1-c}M[a-c+1,2-c,x][/tex]
[tex]y_3=e^{x}M[c-a,c,-x][/tex]
[tex]y_4=x^{1-c}e^{x}M[1-a,2-c,-x][/tex]
[tex]y_5=U[a,c,x][/tex]
[tex]y_6=x^{1-c}U[a-c+1,2-c,x][/tex]
[tex]y_7=e^{x}U[c-a,c,-x][/tex]
[tex]y_8=x^{1-c}e^{x}U[1-a,2-c,-x][/tex]
where ##M[a,c,x]## is the confluent hypergeometric function of the first kind
and ##U[a,c,x]## is the confluent hypergeometric function of the second kind.
I would like to know how to evaluate exactly (step by step) the wronskian of the solutions. For example the ##W(y_1,y_5)## or ##W(y_5,y_7)##.
[tex] W(y_1,y_5) = y_{1} y_{5}^{'} - y_{1}^{'} y_{5} = ? =-\frac{\Gamma(c)}{\Gamma(a)} [/tex]
[tex] W(y_5,y_7) = y_{5} y_{7}^{'} - y_{5}^{'} y_{7} = ? = e^{j\cdot\pi\cdot \text{sign} \left[\Im(x)\right]\cdot (c-a)}[/tex]
The results are given in the referred book but the calculation is missing.
Can anybody help me or suggest a hint? Can anybody offer a article for this problem?
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