## 6 Functional Limits Associated with the Wright Functions
 \tag{77}\]  
In the case of the Laplacian transformation, (77) can be written in the following way:  
\[ \begin{array}{l}\int_{0}^{\infty}e^{-\xi}x_{J_{\
u}}(\
u\,\xi)\,d\xi=\frac{ \
u^{\
u}}{\sqrt{\
u^{2}+\xi^{2}}[\xi+\sqrt{\
u^{2}+\xi^{2}}]^{\
u}}\\ \lim_{\
u\rightarrow\infty}\bigg{\{}{\
u^{\
u}+1}\,\int_{0}^{\infty}\frac{f(\xi \lambda)}{\sqrt{\
u^{2}+\xi^{2}}[\xi+\sqrt{\
u^{2}+\xi^{2}}]^{\
u}}\,d\xi \bigg{\}}=L(1,\lambda)\end{array}  \tag{78}\]Introducing (20) into (78), we have  
\[\lim_{v\to\infty}\left\{v^{v\,+\,1}\,\int_{0}^{\infty}\frac{W_{\alpha,\beta}( \lambda\,\xi)}{\sqrt{v^{2}+\xi^{2}}\,[\xi+\sqrt{v^{2}+\xi^{2}}]^{v}}\,d\xi \right\}=\left.\frac{1}{s}\,E_{\alpha,\beta}(\frac{\lambda}{s})\right|_{s\,=\,1 }=E_{\alpha,\beta}(\lambda) \tag{79}\]  
The same operation performed with (21) gives  
\[ \begin{array}{l}\lim_{v\to\infty}\left\{v^{v\,+\,1}\,\int_{0}^{\infty}\frac{ \,\xi\,W_{\alpha,\beta}(\lambda\,\xi)}{\sqrt{v^{2}+\xi^{2}}\,[\xi+\sqrt{v^{2}+ \xi^{2}}]^{v}}\,d\xi\right\}=\frac{1}{1-\rho}E_{\alpha,\beta}(\frac{\lambda}{1 -\rho})\\ \lim_{v\to\infty}\left\{v^{v\,+\,1}\,\int_{0}^{\infty}\frac{\,e^{-\xi\,W_{ \alpha,\beta}(\lambda\,\xi)}}{\sqrt{v^{2}+\xi^{2}}\,[\xi+\sqrt{v^{2}+\xi^{2}}] ^{v}}\,d\xi\right\}=\frac{1}{1+\rho}E_{\alpha,\beta}(\frac{\lambda}{1+\rho}) \end{array}  \tag{80}\]  
and using (25)  
\[ \begin{array}{l}\lim_{v\to\infty}\left\{v^{v\,+\,1}\,\int_{0}^{\infty}\frac{ \,\xi\,W_{\alpha,\beta}(\lambda\,\xi)}{\sqrt{v^{2}+\xi^{2}}\,[\xi+\sqrt{v^{2}+ \xi^{2}}]^{v}}\,d\xi\right\}=\\ \frac{1}{\alpha\,\lambda}[(\alpha\,\lambda-\beta+1)E_{\alpha,\beta}(\lambda)+ E_{\alpha,\beta\,-\,1}(\lambda)]\end{array}  \tag{81}\]  
The Laplacian transform in (43) leads to  
\[\lim_{v\to\infty}\left\{v^{v\,+\,1}\,\int_{0}^{\infty}\frac{W_{1,\beta\,+\,1} (\lambda\,\xi)}{\sqrt{v^{2}+\xi^{2}}\,[\xi+\sqrt{v^{2}+\xi^{2}}]^{v}}\,d\xi \right\}=\frac{e^{-\,\lambda}}{\lambda^{\beta}\Gamma(\beta)}\gamma(\beta, \lambda)\,=E_{1,\beta\,+\,1}(\lambda) \tag{82}\]  
For integer values of parameters \(\beta\) in (82), the limits of the Wright functions can be represented by simple expressions. For \(\beta=1\), we have  
\[ \begin{array}{l}\lim_{v\to\infty}\left\{v^{v\,+\,1}\,\int_{0}^{\infty}\frac {W_{1,1}(\lambda\,\xi)}{\sqrt{v^{2}+\xi^{2}}\,[\xi+\sqrt{v^{2}+\xi^{2}}]^{v}} \,d\xi\right\}=e^{\lambda}\\ \lim_{v\to\infty}\left\{v^{v\,+\,1}\,\int_{0}^{\infty}\frac{W_{1,1}(-\,\lambda \,\xi)}{\sqrt{v^{2}+\xi^{2}}\,[\xi+\sqrt{v^{2}+\xi^{2}}]^{v}}\,d\xi\right\}=e^ {-\lambda}\end{array}  \tag{83}\]  
and  
\[\lim_{v\to\infty}\left\{v^{v\,+\,1}\,\int_{0}^{\infty}\frac{W_{1,1}(-\, \lambda\,\xi^{2})}{\sqrt{v^{2}+\xi^{2}}\,[\xi+\sqrt{v^{2}+\xi^{2}}]^{v}}\,d \xi\right\}=\frac{1}{\sqrt{1+\lambda^{2}}} \tag{84}\]  
For \(\beta=1\), the corresponding limits are  
\[\lim_{v\to\infty}\left\{v^{v\,+\,1}\,\int_{0}^{\infty}\frac{W_{1,2}(-\, \lambda\,\xi)}{\sqrt{v^{2}+\xi^{2}}\,[\xi+\sqrt{v^{2}+\xi^{2}}]^{v}}\,d\xi \right\}=\frac{2}{\lambda}e^{-\,\lambda}\sinh(\frac{\lambda}{2}) \tag{85}\]  
\[\lim_{v\to\infty}\left\{v^{v\,+\,1}\,\int_{0}^{\infty}\frac{W_{1,2}(\lambda\, \xi)}{\sqrt{v^{2}+\xi^{2}}\,[\xi+\sqrt{v^{2}+\xi^{2}}]^{v}}\,d\xi\right\}= \frac{1}{\lambda}(e^{\lambda}-1) \tag{86}\]  
\[\lim_{v\to\infty}\left\{v^{v\,+\,1}\,\int_{0}^{\infty}\frac{W_{1,2}(-\frac{ \lambda^{2}\,\xi^{2}}{4})}{\sqrt{v^{2}+\xi^{2}}\,[\xi+\sqrt{v^{2}+\xi^{2}}]^ {v}}\,d\xi\right\}=\frac{2}{1+\sqrt{1+\lambda^{2}}} \tag{87}\]  
Similarly, for \(\beta=3\), the functional limit is  
\[ \begin{array}{l}\lim_{v\to\infty}\left\{v^{v\,+\,1}\,\int_{0}^{\infty}\frac {W_{1,3}(-\frac{\lambda^{2}\,\xi^{2}}{4})}{\sqrt{v^{2}+\xi^{2}}\,[\xi+\sqrt{v^{2 }+\xi^{2}}]^{v}}\,d\xi\right\}=\\ \frac{1}{\lambda}\left\{\frac{\lambda}{1+\sqrt{1+\lambda^{2}}}+\frac{1}{3}\left[ \frac{\lambda}{1+\sqrt{1+\lambda^{2}}}\right]^{3}\right\}\end{array}  \tag{88}\]The Laplacian transforms of the Mainardi functions from (58) and (59) are  
\[