suppose , and let be a continuous complex valued function on the pro

1.	suppose , and let be a continuous complex valued function on the product space assume that for each , the function is analytic on define F on Ω by and F is analytic on Ω and
Answer» Correct Question: suppose $\large\rm { [a,b] \subseteq \mathbb{R}}$ , and LET $\large\rm { \varphi}$ be a CONTINUOUS complex valued FUNCTION on the product space $\large\rm { Ω \times [a,b]}$ assume that for each $\large\rm { t \in [a,b]}$ , the function $\large\rm { z \longrightarrow \varphi (z,t)}$ is analytic on $\large\rm{ Ω}$ define F on Ω by $\large\rm { F(z) = \displaystyle\int_{a}^{b} \varphi (z,t) dt, z \in Ω}$ and F is analytic on Ω and $\large\rm { F'(z) = \displaystyle\int_{a}^{b} \frac{\partial \varphi}{\partial z} (z,t) dt , z \in Ω}$ Proof: FIX any disk $\large\rm{ D(z_{0} , r)}$ such that $\large\rm { \bar{D} (z_{0}, r) \subseteq Ω}$ . Then for each $\large\rm { z \in D (z_{0} , r)}$ , $\large\rm { F(z) = \displaystyle\int_{a}^{b} \varphi (z,t) dt}$ $\small\rm{ = \frac{1}{2 \pi i} \displaystyle\int_{a}^{b} \Bigg\lgroup \displaystyle\int_{C(z_{0} ,r)} \frac{\varphi (w,t)}{w-z} dw \Bigg\rgroup dt}$ $\small\rm{ = \frac{1}{2 \pi i} \displaystyle\int_{C(z_{0} , r)} \Bigg\lgroup \displaystyle\int_{a}^{b} \varphi(w,t) dt \bigg\rgroup \frac{1}{w-z} dw}$ $\small\rm { F'(z) = \frac{1}{2 \pi i} \displaystyle\int_{C(z_{0} , r)} \Bigg [ \displaystyle\int_{a}^{b} \varphi(w,t) dt \Bigg ] \frac{1}{(w-z)^{2}} dw}$ $\small\rm { = \displaystyle\int_{a}^{b} \Bigg [ \frac{1}{2 \pi i} \displaystyle\int_{C(z_{0} ,r)} \frac{\varphi (w,t)}{(w-z)^{2}} dw \Bigg ] dt}$ $\large\rm { = \displaystyle\int_{a}^{b} \frac{\partial \varphi}{\partial z} (z,t) dt }$ HENCE proven.

suppose , and let be a continuous complex valued function on the product space assume that for each , the function is analytic on define F on Ω by and F is analytic on Ω and

Answer»

Correct Question:

suppose $\large\rm { [a,b] \subseteq \mathbb{R}}$ , and LET $\large\rm { \varphi}$ be a CONTINUOUS complex valued FUNCTION on the product space $\large\rm { Ω \times [a,b]}$ assume that for each $\large\rm { t \in [a,b]}$ , the function $\large\rm { z \longrightarrow \varphi (z,t)}$ is analytic on $\large\rm{ Ω}$ define F on Ω by $\large\rm { F(z) = \displaystyle\int_{a}^{b} \varphi (z,t) dt, z \in Ω}$ and F is analytic on Ω and $\large\rm { F'(z) = \displaystyle\int_{a}^{b} \frac{\partial \varphi}{\partial z} (z,t) dt , z \in Ω}$

Proof:

FIX any disk $\large\rm{ D(z_{0} , r)}$ such that $\large\rm { \bar{D} (z_{0}, r) \subseteq Ω}$ . Then for each $\large\rm { z \in D (z_{0} , r)}$ ,

$\large\rm { F(z) = \displaystyle\int_{a}^{b} \varphi (z,t) dt}$

$\small\rm{ = \frac{1}{2 \pi i} \displaystyle\int_{a}^{b} \Bigg\lgroup \displaystyle\int_{C(z_{0} ,r)} \frac{\varphi (w,t)}{w-z} dw \Bigg\rgroup dt}$

$\small\rm{ = \frac{1}{2 \pi i} \displaystyle\int_{C(z_{0} , r)} \Bigg\lgroup \displaystyle\int_{a}^{b} \varphi(w,t) dt \bigg\rgroup \frac{1}{w-z} dw}$

$\small\rm { F'(z) = \frac{1}{2 \pi i} \displaystyle\int_{C(z_{0} , r)} \Bigg [ \displaystyle\int_{a}^{b} \varphi(w,t) dt \Bigg ] \frac{1}{(w-z)^{2}} dw}$

$\small\rm { = \displaystyle\int_{a}^{b} \Bigg [ \frac{1}{2 \pi i} \displaystyle\int_{C(z_{0} ,r)} \frac{\varphi (w,t)}{(w-z)^{2}} dw \Bigg ] dt}$

$\large\rm { = \displaystyle\int_{a}^{b} \frac{\partial \varphi}{\partial z} (z,t) dt }$

HENCE proven.

suppose , and let be a continuous complex valued function on the product space assume that for each , the function is analytic on define F on Ω by and F is analytic on Ω and

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suppose , and let be a continuous complex valued function on the product space assume that for each , the function is analytic on define F on Ω by and F is analytic on Ω and ​

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suppose , and let be a continuous complex valued function on the product space assume that for each , the function is analytic on define F on Ω by and F is analytic on Ω and