Question Solved1 Answer by complex Use the theorem in Sec. 23 to show that \( f^{\prime}(z) \) and its derivative \( f^{\prime \prime}(z) \) exist everywhere, and find \( f^{\prime \prime}(z) \) when (a) \( f(z)=i z+2 \); (b) \( f(z)=e^{-x} e^{-i y} \); (c) \( f(z)=z^{3} \); (d) \( f(z)=\cos x \cosh y-i \sin x \sinh y \). Ans. (b) \( f^{\prime \prime}(z)=f(z) \); (d) \( f^{\prime \prime}(z)=-f(z) \).

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Transcribed Image Text: Use the theorem in Sec. 23 to show that \( f^{\prime}(z) \) and its derivative \( f^{\prime \prime}(z) \) exist everywhere, and find \( f^{\prime \prime}(z) \) when (a) \( f(z)=i z+2 \); (b) \( f(z)=e^{-x} e^{-i y} \); (c) \( f(z)=z^{3} \); (d) \( f(z)=\cos x \cosh y-i \sin x \sinh y \). Ans. (b) \( f^{\prime \prime}(z)=f(z) \); (d) \( f^{\prime \prime}(z)=-f(z) \).
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Transcribed Image Text: Use the theorem in Sec. 23 to show that \( f^{\prime}(z) \) and its derivative \( f^{\prime \prime}(z) \) exist everywhere, and find \( f^{\prime \prime}(z) \) when (a) \( f(z)=i z+2 \); (b) \( f(z)=e^{-x} e^{-i y} \); (c) \( f(z)=z^{3} \); (d) \( f(z)=\cos x \cosh y-i \sin x \sinh y \). Ans. (b) \( f^{\prime \prime}(z)=f(z) \); (d) \( f^{\prime \prime}(z)=-f(z) \).
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Step1/4mjx-container[jax="CHTML"]{line-height: 0;}mjx-container [space="3"]{margin-left: .222em;}mjx-container [space="4"]{margin-left: .278em;}mjx-container [size="s"]{font-size: 70.7%;}mjx-itable{display: inline-table;}mjx-assistive-mml{position: absolute !important; top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); padding: 1px 0px 0px 0px !important; border: 0px !important; display: block !important; width: auto !important; overflow: hidden !important; -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none;}mjx-math{display: inline-block; text-align: left; line-height: 0; text-indent: 0; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; border-collapse: collapse; word-wrap: normal; word-spacing: normal; white-space: nowrap; direction: ltr; padding: 1px 0;}mjx-mtable{display: inline-block; text-align: center; vertical-align: .25em; position: relative; box-sizing: border-box; border-spacing: 0; border-collapse: collapse;}mjx-table{display: inline-block; vertical-align: -.5ex; box-sizing: border-box;}mjx-table > mjx-itable{vertical-align: middle; text-align: left; box-sizing: border-box;}mjx-mtr{display: table-row; text-align: left;}mjx-mtd{display: table-cell; text-align: center; padding: .215em .4em;}mjx-mtd:first-child{padding-left: 0;}mjx-mtd:last-child{padding-right: 0;}mjx-mtable > * > mjx-itable > *:first-child > mjx-mtd{padding-top: 0;}mjx-mtable > * > mjx-itable > *:last-child > mjx-mtd{padding-bottom: 0;}mjx-tstrut{display: inline-block; height: 1em; vertical-align: -.25em;}mjx-mi{display: inline-block; text-align: left;}mjx-c{display: inline-block;}mjx-mrow{display: inline-block; text-align: left;}mjx-mo{display: inline-block; text-align: left;}mjx-mn{display: inline-block; text-align: left;}mjx-msup{display: inline-block; text-align: left;}mjx-c::before{display: block; width: 0;}.MJX-TEX{font-family: MJXZERO, MJXTEX;}@font-face{font-family: MJXZERO; src: url("https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_Zero.woff") format("woff");}@font-face{font-family: MJXTEX; src: url("https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_Main-Regular.woff") format("woff");}mjx-c.mjx-c66::before{padding: 0.705em 0.372em 0 0; content: "f";}mjx-c.mjx-c28::before{padding: 0.75em 0.389em 0.25em 0; content: "(";}mjx-c.mjx-c7A::before{padding: 0.431em 0.444em 0 0; content: "z";}mjx-c.mjx-c29::before{padding: 0.75em 0.389em 0.25em 0; content: ")";}mjx-c.mjx-c3D::before{padding: 0.583em 0.778em 0.082em 0; content: "=";}mjx-c.mjx-c69::before{padding: 0.669em 0.278em 0 0; content: "i";}mjx-c.mjx-c2B::before{padding: 0.583em 0.778em 0.082em 0; content: "+";}mjx-c.mjx-c32::before{padding: 0.666em 0.5em 0 0; content: "2";}mjx-c.mjx-c2032::before{padding: 0.56em 0.275em 0 0; content: "\2032";}mjx-c.mjx-c2033::before{padding: 0.56em 0.55em 0 0; content: "′′";}mjx-c.mjx-c30::before{padding: 0.666em 0.5em 0.022em 0; content: "0";}a) f(z)=iz+2Differentiating above eq with respect to z.f′(z)=iSince there is not any pole. Hence f′(z)exists everywhere.Differentiating above eq again with respect to z.f″(z)=0Hence f″(z)exists everywhere.Explanation:If there exists a complex number at the denominator then we say that it is a pole. If a pole exists then the function and its derivatives are not defined around and in the neighborhood of that point.Differentiating the given function we find that there is not any pole hence the function exists everywhere.Step2/4mjx-container[jax="CHTML"]{line-height: 0;}mjx-container [space="4"]{margin-left: .278em;}mjx-container [size="s"]{font-size: 70.7%;}mjx-itable{display: inline-table;}mjx-assistive-mml{position: absolute !important; top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); padding: 1px 0px 0px 0px !important; border: 0px !important; display: block !important; width: auto !important; overflow: hidden !important; -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none;}mjx-math{display: inline-block; text-align: left; line-height: 0; text-indent: 0; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; border-collapse: collapse; word-wrap: normal; word-spacing: normal; white-space: nowrap; direction: ltr; padding: 1px 0;}mjx-mtable{display: inline-block; text-align: center; vertical-align: .25em; position: relative; box-sizing: border-box; border-spacing: 0; border-collapse: collapse;}mjx-table{display: inline-block; vertical-align: -.5ex; box-sizing: border-box;}mjx-table > mjx-itable{vertical-align: middle; text-align: left; box-sizing: border-box;}mjx-mtr{display: table-row; text-align: left;}mjx-mtd{display: table-cell; text-align: center; padding: .215em .4em;}mjx-mtd:first-child{padding-left: 0;}mjx-mtd:last-child{padding-right: 0;}mjx-mtable > * > mjx-itable > *:first-child > mjx-mtd{padding-top: 0;}mjx-mtable > * > mjx-itable > *:last-child > mjx-mtd{padding-bottom: 0;}mjx-tstrut{display: inline-block; height: 1em; vertical-align: -.25em;}mjx-mi{display: inline-block; text-align: left;}mjx-c{display: inline-block;}mjx-mrow{display: inline-block; text-align: left;}mjx-mo{display: inline-block; text-align: left;}mjx-msup{display: inline-block; text-align: left;}mjx-c::before{display: block; width: 0;}.MJX-TEX{font-family: MJXZERO, MJXTEX;}@font-face{font-family: MJXZERO; src: url("https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_Zero.woff") format("woff");}@font-face{font-family: MJXTEX; src: url("https://cdn.jsdelivr.net/npm/mathjax@3/es5/output/chtml/fonts/woff-v2/MathJax_Main-Regular.woff") format("woff");}mjx-c.mjx-c66::before{padding: 0.705em 0.372em 0 0; content: "f";}mjx-c.mjx-c28::before{padding: 0.75em 0.389em 0.25em 0; content: "(";}mjx-c.mjx-c7A::before{padding: 0.431em 0.444em 0 0; content: "z";}mjx-c.mjx-c29::before{padding: 0.75em 0.389em 0.25em 0; content: ")";}mjx-c.mjx-c3D::before{padding: 0.583em 0.778em 0.082em 0; content: "=";}mjx-c.mjx-c65::before{padding: 0.448em 0.444em 0.011em 0; content: "e";}mjx-c.mjx-c2212::before{padding: 0.583em 0.778em 0.082em 0; content: "\2212";}mjx-c.mjx-c78::before{padding: 0.431em 0.528em 0 0; content: "x";}mjx-c.mjx-c69::before{padding: 0.669em 0.278em 0 0; content: "i";}mjx-c.mjx-c79::before{padding: 0.431em 0.528em 0.204em 0; content: "y";}mjx-c.mjx-c2B::before{padding: 0.583em 0.778em 0.082em 0; content: "+";}mjx-c.mjx-c2032::before{padding: 0.56em 0.275em 0 0; content: "\2032";}mjx-c.mjx-c2033::before{padding: 0.56em 0.55em 0 0; content: "′′";}b) f(z)=e−xe−(iy)f(z)=e−(x+iy)f(z)=e−zDifferentiating the above equation we get f′(z)=−e−zSince there is not any pole hence it exists everywhere.Differentiating again the above equation we get f″(z)=e−zSince there is not any pole hence it exists everywhere.Explanation:If there exists a complex number at the denominator then we say that it is a pole. If a pole exists then the function and its derivatives are not defined around and in the neighborhood of that point.Differentiating the given function we find that there is not any pole hence the function exists everywhere.Step3/4mjx-container[jax="CHTML"]{line-height: 0;}mjx-container [space="4"]{margin-left: .278em;}mjx-container [size="s"]{font-size: 70.7%;}mjx-itable{display: inline-table;}mjx-assistive-mml{position: absolute !important; top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); padding: 1px 0px 0px 0px !important; border: 0px !important; display: block !important; wi ... See the full answer