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

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

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

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

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

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

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

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

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

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

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

• A) \[\cosh z\]
• B) \[\tanh z\]
• C) \[{\csc ^2}z\]
• D) \[\sec {h^2}z\]

{\text{For any complex number z, cosh }}z{\text{ = _____________}}{\text{.}}

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

\[\begin{gathered} {\text{For z = x + i y, we have}} \hfill \\\ {\text{sin z = ____________}}{\text{.}} \hfill \\\\ \end{gathered} \]

• A) \[\sin x\cos y + i\cos x\sin y\]
• B) \[\cos x\sin y + i\sin x\cos y\]
• C) \[\cos x\sinh y + i\sin x\cosh y\]
• D) \[\sin x\cosh y + i\cos x\sinh y\]

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

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

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

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

\[{\text{For the power series }}\sum\limits_{n = 0}^\infty {(n + 1){Z^n}} {\text{ is equal to _________}}{\text{.}}\]

• A) \[\frac{1}{{{{(1 - Z)}^2}}}\]
• B) \[\frac{1}{{{Z^2}}}\]
• C) \[\frac{1}{{{{(1 + Z)}^2}}}\]
• D) \[\frac{1}{Z}\]

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

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

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

• A) \[\sec {h^2}z\]
• B) \[\sin z\]
• C) \[\tanh z\]
• D) \[\sinh z\]

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

• A) z
• B) 1/z
• C) 1/(1+z)
• D) iz

\[{\text{For any complex number z, tanh }}z{\text{ = ____________}}{\text{.}}\]

• 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) \[\frac{{{e^z} - {e^{ - z}}}}{{{e^z} + {e^{ - z}}}}\]

\[\begin{gathered} {\text{For z = x + i y, we have}} \hfill \ {\text{cos z = ___________}}{\text{.}} \hfill \\\ \end{gathered} \]

• A) \[\sin x\sinh y + i\cos x\cosh y\]
• B) \[\cos x\cosh y - i\sin x\sinh y\]
• C) \[\sin x\cosh y + i\cos x\sinh y\]
• D) \[\cos x\cosh y + i\sin x\sinh y\]

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

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