Sec 35-36 The Power function, 37 Trig functions Flashcards

1
Q

$z^c$ called

A

the multiple valued power function

$z^C$ for $z \not \in 0$ and $c \in \mathbb{C}$

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2
Q

$z^c$ =

A

$e^{c \ logz}$

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3
Q

$\frac{1}{z^c} = $

A

$z^{-c}$

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4
Q

$\frac{d}{dz} (z^c_{\alpha}) = $

A

$c \ z^{c-1}_{\alpha} $

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5
Q

$z^c_{\alpha} = $

A

$e^{c \ log_{\alpha}z}$

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6
Q

Principle values of $z^c$, the multiple valued complex power function is

A

$e^{c\ Log z}$

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7
Q

how is $\sin{z}$ defined for complex z

A

$\sin{z} = \frac{e^{iz} - e^{-iz}}{2i}$

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8
Q

how is $\cos{z}$ defined for complex z

A

$\cos{z} = \frac{e^{iz} + e^{-iz}}{2}$

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9
Q

$sin(z_1 + z_2)$

A

As real case

=$sin(z_1) cos(z_2) + cos(z_1)sin(z_2) $

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10
Q

$cos(z_1 + z_2)$

A

As real case

=$cos(z_1) cos(z_2) - sin(z_1)sin(z_2) $

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11
Q

how is $\sinh{z}$ defined for complex z

same as reals

A

$\sinh{z} = \frac{e^{z} - e^{-z}}{2}$

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12
Q

how is $\cosh{z}$ defined for complex z

same as reals

A

$\cosh{z} = \frac{e^{z} + e^{-z}}{2}$

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