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complex_exponents_with_positive_real_bases [2015/01/12 18:43] nikolaj |
complex_exponents_with_positive_real_bases [2015/04/15 14:12] nikolaj |
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| | @#FFBB00: definiendum | @#FFBB00: $ z\mapsto b^z := \mathrm{exp}(z\cdot \mathrm{ln}(b)) $ | | | @#FFBB00: definiendum | @#FFBB00: $ z\mapsto b^z := \mathrm{exp}(z\cdot \mathrm{ln}(b)) $ | | ||
| - | ==== Discussion ==== | + | ----- |
| - | ==== Parents ==== | + | === Discussion === |
| + | The identity | ||
| + | |||
| + | $b^{x_1+x_2}=b^{x_1}\cdot a^{x_2}$, | ||
| + | |||
| + | says that exponentiation is a (the) homomorphism between $+$ and $\cdot$. | ||
| + | |||
| + | The combinatorial manifestation, e.g. formulated in for $B,X_1,\dots\in\bf{Set}$, is | ||
| + | |||
| + | $B^{\coprod_{j\in J}X_j}\cong\prod_{j\in J} B^{X_j}$ | ||
| + | |||
| + | ----- | ||
| === Context === | === Context === | ||
| [[Natural logarithm of real numbers]] | [[Natural logarithm of real numbers]] | ||
