Why is the infinite series used in constructing complex valued functions?

• A) To simplify the calculations
• B) To make them shorter
• C) To improve convergence
• D) To make them infinitely long

Period of the function ez is _____________ degree.

• A) 360
• B) 180
• C) 0
• D) 90

Under what condition does an infinite power series define a function?

• A) When the series has an odd number of terms.
• B) When the series is convergent.
• C) When the series is divergent.
• D) When the series has a constant term.

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

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

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

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

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

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

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

We take the derivative of principal branch of zc because its principal branch is ______________ function.

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

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

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

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

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

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

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

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

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

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QCeAAAAAAAAAYEAngAAAAAAAAGBAJ4AAAAAAAABgQCeAAAAAAAAAYEAngAAAAAAAAGBAJ4AAAAAAAABgQCeAAAAAAAAAYD0X97K/s//R2gCwAAAABJRU5ErkJggg== .

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

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

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

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

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

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

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

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

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

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

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

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

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

• 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) $$\frac{{{e^{iz}} - {e^{ - iz}}}}{{{e^{iz}} + {e^{ - iz}}}}$$