General solution to complex number to complex power in complex form
Given the form:
$(a+bi)^{(c+di)}$
Does there exist a generalized solution for the principle branch where:
$(a+bi)^{(c+di)} = (e+fi)$
I ask this because addition and multiplication (with subtraction and division) have generalized solutions:
$(a+bi) + (c+di) = (e+fi)$
$e=a+b$
$f=b+d$
and for multiplication:
$(a+bi)*(c+di) = (e+fi)$
$e=a*c-b*d$
$f=b*c+a*d$
I also realize that:
$(a+bi)^{(c+di)} = e^{(c+di)log(a+bi)}$
and that this can be converted to polar form to solve this problem; however, I'm not sure how to reduce this to a complex form afterwards.
complex-numbers exponential-function polar-coordinates
asked Nov 29 at 18:49
Michael Choi
156
add a comment |
Given the form:
$(a+bi)^{(c+di)}$
Does there exist a generalized solution for the principle branch where:
$(a+bi)^{(c+di)} = (e+fi)$
I ask this because addition and multiplication (with subtraction and division) have generalized solutions:
$(a+bi) + (c+di) = (e+fi)$
$e=a+b$
$f=b+d$
and for multiplication:
$(a+bi)*(c+di) = (e+fi)$
$e=a*c-b*d$
$f=b*c+a*d$
I also realize that:
$(a+bi)^{(c+di)} = e^{(c+di)log(a+bi)}$
and that this can be converted to polar form to solve this problem; however, I'm not sure how to reduce this to a complex form afterwards.
complex-numbers exponential-function polar-coordinates
asked Nov 29 at 18:49
Michael Choi
156
add a comment |
Given the form:
$(a+bi)^{(c+di)}$
Does there exist a generalized solution for the principle branch where:
$(a+bi)^{(c+di)} = (e+fi)$
I ask this because addition and multiplication (with subtraction and division) have generalized solutions:
$(a+bi) + (c+di) = (e+fi)$
$e=a+b$
$f=b+d$
and for multiplication:
$(a+bi)*(c+di) = (e+fi)$
$e=a*c-b*d$
$f=b*c+a*d$
I also realize that:
$(a+bi)^{(c+di)} = e^{(c+di)log(a+bi)}$
and that this can be converted to polar form to solve this problem; however, I'm not sure how to reduce this to a complex form afterwards.
complex-numbers exponential-function polar-coordinates
asked Nov 29 at 18:49
Michael Choi
156
Given the form:
$(a+bi)^{(c+di)}$
Does there exist a generalized solution for the principle branch where:
$(a+bi)^{(c+di)} = (e+fi)$
I ask this because addition and multiplication (with subtraction and division) have generalized solutions:
$(a+bi) + (c+di) = (e+fi)$
$e=a+b$
$f=b+d$
and for multiplication:
$(a+bi)*(c+di) = (e+fi)$
$e=a*c-b*d$
$f=b*c+a*d$
I also realize that:
$(a+bi)^{(c+di)} = e^{(c+di)log(a+bi)}$
and that this can be converted to polar form to solve this problem; however, I'm not sure how to reduce this to a complex form afterwards.
complex-numbers exponential-function polar-coordinates
complex-numbers exponential-function polar-coordinates
asked Nov 29 at 18:49
Michael Choi
156
asked Nov 29 at 18:49
Michael Choi
156
asked Nov 29 at 18:49
Michael Choi
156
asked Nov 29 at 18:49
Michael Choi
156
asked Nov 29 at 18:49
Michael Choi
156
156
add a comment |
add a comment |
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