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

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

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

{\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

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

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

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

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

{\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^z} - {e^{ - z}}}}{{{e^z} + {e^{ - z}}}}
• D) \frac{{{e^{iz}} - {e^{ - iz}}}}{{{e^{iz}} + {e^{ - iz}}}}

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

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

{\text{For any complex number z, csch }}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}}}}

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

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

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

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

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

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

Range of principle argument of Log(z) is __________.

• A) ( - \pi ,\pi ]
• B) ( - \pi ,\pi )
• C) [ - \pi ,\pi ]
• D) [ - \pi ,\pi )

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

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

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

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

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• A) single valued
• B) multivalued
• C)
• D)

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• A) finite values
• B) infinite values
• C)
• D)