$${\text{For any complex number z, csch }}z{\text{ = __________}}{\text{.}}$$

• A) $$\frac{2}{{{e^z} + {e^{ - z}}}}$$
• B) $$\frac{{{e^z} - {e^{ - z}}}}{2}$$
• C) $$\frac{{{e^z} + {e^{ - z}}}}{2}$$
• D) $$\frac{2}{{{e^z} - {e^{ - z}}}}$$

$${\text{For any complex number z, }}\sinh z{\text{ = ___________}}{\text{.}}$$

• A) $$\frac{{{e^{iz}} + {e^{ - iz}}}}{2}$$
• B) $$\frac{{{e^z} - {e^{ - z}}}}{2}$$
• C) $$\frac{{{e^z} + {e^{ - z}}}}{2}$$
• D) $$\frac{{{e^{iz}} - {e^{ - iz}}}}{{2i}}$$

data:image/png;base64,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 .

• A) 0.20788
• B) 0.30788
• C) 0.40788
• D) 0.10788

data:image/png;base64,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 .

• A) data:image/png;base64,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 .
• B) data:image/png;base64,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 .
• C) data:image/png;base64,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 .
• D) data:image/png;base64,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 .

\frac{d}{{dz}}(lo{g_\alpha }z) = \_\_\_\_\_\_\_\_\_\_,{\text{ for z = r }}{{\text{e}}^{i\theta }},{\text{ }}\alpha < \theta < \alpha + 2\pi .

• A) 1/(1+z)
• B) iz
• C) 1/z
• D) z

{\text{For any complex number z, }}\frac{d}{{dz}}{\text{(cot z) = _________}}{\text{.}}

• A) \tan z\sec z
• B) - \sec z\tan z
• C) - {\csc ^2}z
• D) - {\sec ^2}z

Range of Log(z) is ___________.

• A) \ln |z| + iarg(z)
• B) \ln |z|
• C) \ln |z| + arg(z)
• D) \ln (z) + arg(z)

\[{\text{For any complex number z, }}\sinh z{\text{ = ___________}}{\text{.}}\]

• A) \[\frac{{{e^{iz}} - {e^{ - iz}}}}{{2i}}\]
• B) \[\frac{{{e^z} + {e^{ - z}}}}{2}\]
• C) \[\frac{{{e^z} - {e^{ - z}}}}{2}\]
• D) \[\frac{{{e^{iz}} + {e^{ - iz}}}}{2}\]

The complex exponential function w = ez is one-to-one if we only consider ___________

• A) the argument
• B) the principal argument
• C)
• D)

\[{\text{For any complex number z, }}\frac{d}{{dz}}{\text{(sec z) = __________}}{\text{.}}\]

• A) \[ - \sec z\tan z\]
• B) \[-\cot z\sec z\]
• C) \[\cot z\sec z\]
• D) \[\tan z\sec z\]

The function ez is an entire function.

• A) False
• B) True
• C)
• D)

{\text{For any complex number z, }}\frac{d}{{dz}}{\text{(csc z) = ____________}}{\text{.}}

• A) - \cot z\csc z
• B) \tan z\sec z
• C) \cot z\sec z
• D) \cot z\operatorname{cscz}

{\text{For any complex number z, }}\frac{d}{{dz}}{\text{(sec z) = __________}}{\text{.}}

• A) \cot z\sec z
• B) -\cot z\sec z
• C) - \sec z\tan z
• D) \tan z\sec z

{\text{For any complex number z, sech }}z{\text{ = ___________}}{\text{.}}

• A) \frac{{{e^z} + {e^{ - z}}}}{2}
• B) \frac{2}{{{e^z} + {e^{ - z}}}}
• C) \frac{{{e^z} - {e^{ - z}}}}{2}
• D) \frac{2}{{{e^z} - {e^{ - z}}}}

\[{\text{For any complex number z, coth }}z{\text{ = _____________}}{\text{.}}\]

• A) \[\frac{{{e^z} - {e^{ - z}}}}{{{e^z} + {e^{ - z}}}}\]
• B) \[\frac{{{e^{iz}} + {e^{ - iz}}}}{{{e^{iz}} - {e^{ - iz}}}}\]
• C) \[\frac{{{e^{iz}} - {e^{ - iz}}}}{{{e^{iz}} + {e^{ - iz}}}}\]
• D) \[\frac{{{e^z} + {e^{ - z}}}}{{{e^z} - {e^{ - z}}}}\]

Complex exponential function is ___________.

• A) one to one function
• B) not one to one function
• C)
• D)

{\text{For any complex number z, }}\sinh z{\text{ = ___________}}{\text{.}}

• A) \frac{{{e^z} - {e^{ - z}}}}{2}
• B) \frac{{{e^{iz}} + {e^{ - iz}}}}{2}
• C) \frac{{{e^z} + {e^{ - z}}}}{2}
• D) \frac{{{e^{iz}} - {e^{ - iz}}}}{{2i}}

{\text{For the power series }}\sum\limits_{n = 0}^\infty {{Z^n}} {\text{ is equal to __________}{\text{.}}

• A) 1/(1-Z)
• B) 1/(1+Z)
• C) 1/Z
• D) Z

{\text{For any complex number z, }}\frac{d}{{dz}}{\text{sech }}z{\text{ = ___________}}{\text{.}}

• A) \sinh z
• B) - \csc {h^2}z
• C) \sec {h^2}z
• D) - \sec hz\tanh z

{\text{For any complex number z, }}\frac{d}{{dz}}{\text{coth }}z{\text{ = ____________}}{\text{.}}

• A) - \csc hz\coth z
• B) - \csc {h^2}z
• C) \sinh z
• D) \sec {h^2}z